Welcome to the linear independence calculator, where we'll learn how to check if you're dealing with linearly independent vectors or not. The world we live in is 3-dimensional, so the vectors will have three coordinates. To check linear dependence, we'll translate our problem from the language of vectors into the language of matrices (arrays of numbers).

We have 3 vectors with 3 coordinates each, so we start by telling the calculator that fact by choosing the appropriate options under "number of vectors" and "number of coordinates." Then, we input the vectors.

For example, let's say that we're given three vectors in a 2-dimensional space (with two coordinates): v = (a₁, a₂), w = (b₁, b₂), and u = (c₁, c₂). We write their coordinates as one big matrix with each row (or column, it doesn't matter) corresponding to one of the vectors.

The rank of the matrix is equal to the maximal number of linearly independent vectors among v, w, and u. The dimensions of the square are found by calculating the distance between various corner points. Recall that we can find the distance between any two points if we know their coordinates.

The formula used to find the distance is: Distance = sqrt[(x₂ - x₁)² + (y₂ - y₁)²]. To calculate the slope of the sides of the triangle, the formula used is: Slope = (y₂ - y₁) / (x₂ - x₁).

The formula used to find the height is: h = 2 * (K / a), where K is the area of the triangle. The formula used is: Centroid x = 1/3(Ax + Bx + Cx) Centroid y = 1/3(Ay + By + Cy).

The formula used to find the radius is: R = (abc) / (4K), where K is the area of the triangle. The formula used is: D = 2(Bx*Cy - By*Cx) Ux = Cy(Bx² + By²) - By(Cx² + Cy²) / D Uy = Bx(Cx² + Cy²) - Cx(Bx² + By²) / D.

The formula used to find the height is: ha = 2 * (K / a), where K is the area of the triangle. The formula used to find the height is: hc = 2 * (K / c), where K is the area of the triangle.

The formula used to find the height is: hb = 2 * (K / b), where K is the area of the triangle. The formula used is: P = a + b + c. This is the perimeter (P) of the solved triangle.

The formula used for finding the length of the line is: mc = (1/2)sqrt[2a² + 2b² - c²]. The formula used for finding the length of the line is: ma = (1/2)sqrt[2c² + 2b² - a²].

The formula used for finding the length of the line is: mb = (1/2)sqrt[2a² + 2c² - b²]. This is the length of side a of the solved triangle.

This is the length of side b of the solved triangle. This is the length of side c of the solved triangle.

This is the semi-perimeter (s) of the solved triangle. This is the area (K) of the solved triangle.

This is the radius (R) of the circumscribed circle of the solved triangle. This is the radius (r) of the inscribed circle of the solved triangle.

This is the angle of vertex A of the solved triangle. This is the angle of vertex B of the solved triangle.

This is the angle of vertex C of the solved triangle. The formula used to find the height is: h = (2 * K) / a, where K is the area of the triangle.

The formula used to find the height is: h = (2 * K) / b, where K is the area of the triangle.

The formula used to find the height is: h = (2 * K) / c, where K is the area of the triangle.

The formula used to find the radius is: r = sqrt[((s-a)*(s-b)*(s-c))/s], where s is the semi-perimeter of the triangle.

The formula used to find the radius is: R = (a*b*c) / (4*K), where K is the area of the triangle.

The formula used is: D = 2(Bx*Cy - By*Cx) Ux = Cy(Bx² + By²) - By(Cx² + Cy²) / D Uy = Bx(Cx² + Cy²) - Cx(Bx² + By²) / D.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the type of triangle based on the entered and computed triangle values.

The inscribed circle is the largest circle that will fit within the triangle.

The circumscribed circle is a circle that passes through all three vertices of a triangle.

The formula used is K = (base x height) / 2.

The formula used is P = a + b + c.

The formula used for finding the length of the line is: ma = (1/2)sqrt[2c² + 2b² - a²].

The formula used for finding the length of the line is: mb = (1/2)sqrt[2a² + 2c² - b²].

The formula used for finding the length of the line is: mc = (1/2)sqrt[2a² + 2b² - c²].

This is the length of the median (ma), which is the line that runs from vertex A to the mid-point of side a (the opposite side).

This is the length of the median (mb), which is the line that runs from vertex B to the mid-point of side b (the opposite side).

This is the length of the median (mc), which is the line that runs from vertex C to the mid-point of side c (the opposite side).

This is the height (h) of the triangle using side a as the base.

This is the height (h) of the triangle using side b as the base.

This is the height (h) of the triangle using side c as the base.

This is the angle of vertex A of the solved triangle.

This is the angle of vertex B of the solved triangle.

This is the angle of vertex C of the solved triangle.

To save changes to previously saved entries, simply tap the Save button.

If you gave the calculator less than "A", please help me improve the calculator by telling me what I would need to do for it to earn an "A".

Clicking the "Reset" button will restore the calculator to its default settings.

Please select and "Clear" any data records you no longer need.

Note that my expertise is in creating online calculators, not necessarily in all of the subject areas they cover.

If it's not filled in, please enter the web address of the calculator as displayed in the location field at the top of the browser window (www.free-online-calculator-use.com/____.html).

This field should already be filled in if you are using a newer web browser with javascript turned on.

A Data Record is a set of calculator entries that are stored in your web browser's Local Storage.

If you have a question about the calculator's operation, please enter your question, your first name, and a valid email address.

The formula used to find the height is: h = (2 * K) / c, where K is the area of the triangle.

The formula used to find the radius is: R = (a*b*c) / (4*K), where K is the area of the triangle.

The formula used to find the radius is: r = sqrt[((s-a)*(s-b)*(s-c))/s], where s is the semi-perimeter of the triangle.

The formula used is: D = 2(Bx*Cy - By*Cx) Ux = Cy(Bx² + By²) - By(Cx² + Cy²) / D Uy = Bx(Cx² + Cy²) - Cx(Bx² + By²) / D.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the type of triangle based on the entered and computed triangle values.

The inscribed circle is the largest circle that will fit within the triangle.

The circumscribed circle is a circle that passes through all three vertices of a triangle.

The formula used is K = (base x height) / 2.

The formula used is P = a + b + c.

The formula used for finding the length of the line is: ma = (1/2)sqrt[2c² + 2b² - a²].

The formula used for finding the length of the line is: mb = (1/2)sqrt[2a² + 2c² - b²].

The formula used for finding the length of the line is: mc = (1/2)sqrt[2a² + 2b² - c²].

This is the length of the median (ma), which is the line that runs from vertex A to the mid-point of side a (the opposite side).

This is the length of the median (mb), which is the line that runs from vertex B to the mid-point of side b (the opposite side).

This is the length of the median (mc), which is the line that runs from vertex C to the mid-point of side c (the opposite side).

This is the height (h) of the triangle using side a as the base.

This is the height (h) of the triangle using side b as the base.

This is the height (h) of the triangle using side c as the base.

This is the angle of vertex A of the solved triangle.

This is the angle of vertex B of the solved triangle.

This is the angle of vertex C of the solved triangle.

To save changes to previously saved entries, simply tap the Save button.

If you gave the calculator less than "A", please help me improve the calculator by telling me what I would need to do for it to earn an "A".

Clicking the "Reset" button will restore the calculator to its default settings.

Please select and "Clear" any data records you no longer need.

Note that my expertise is in creating online calculators, not necessarily in all of the subject areas they cover.

If it's not filled in, please enter the web address of the calculator as displayed in the location field at the top of the browser window (www.free-online-calculator-use.com/____.html).

This field should already be filled in if you are using a newer web browser with javascript turned on.

A Data Record is a set of calculator entries that are stored in your web browser's Local Storage.

If you have a question about the calculator's operation, please enter your question, your first name, and a valid email address.

The formula used to find the height is: h = (2 * K) / c, where K is the area of the triangle.

The formula used to find the radius is: R = (a*b*c) / (4*K), where K is the area of the triangle.

The formula used to find the radius is: r = sqrt[((s-a)*(s-b)*(s-c))/s], where s is the semi-perimeter of the triangle.

The formula used is: D = 2(Bx*Cy - By*Cx) Ux = Cy(Bx² + By²) - By(Cx² + Cy²) / D Uy = Bx(Cx² + Cy²) - Cx(Bx² + By²) / D.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the type of triangle based on the entered and computed triangle values.

The inscribed circle is the largest circle that will fit within the triangle.

The circumscribed circle is a circle that passes through all three vertices of a triangle.

The formula used is K = (base x height) / 2.

The formula used is P = a + b + c.

The formula used for finding the length of the line is: ma = (1/2)sqrt[2c² + 2b² - a²].

The formula used for finding the length of the line is: mb = (1/2)sqrt[2a² + 2c² - b²].

The formula used for finding the length of the line is: mc = (1/2)sqrt[2a² + 2b² - c²].

This is the length of the median (ma), which is the line that runs from vertex A to the mid-point of side a (the opposite side).

This is the length of the median (mb), which is the line that runs from vertex B to the mid-point of side b (the opposite side).

This is the length of the median (mc), which is the line that runs from vertex C to the mid-point of side c (the opposite side).

This is the height (h) of the triangle using side a as the base.

This is the height (h) of the triangle using side b as the base.

This is the height (h) of the triangle using side c as the base.

This is the angle of vertex A of the solved triangle.

This is the angle of vertex B of the solved triangle.

This is the angle of vertex C of the solved triangle.

To save changes to previously saved entries, simply tap the Save button.

If you gave the calculator less than "A", please help me improve the calculator by telling me what I would need to do for it to earn an "A".

Clicking the "Reset" button will restore the calculator to its default settings.

Please select and "Clear" any data records you no longer need.

Note that my expertise is in creating online calculators, not necessarily in all of the subject areas they cover.

If it's not filled in, please enter the web address of the calculator as displayed in the location field at the top of the browser window (www.free-online-calculator-use.com/____.html).

This field should already be filled in if you are using a newer web browser with javascript turned on.

A Data Record is a set of calculator entries that are stored in your web browser's Local Storage.

If you have a question about the calculator's operation, please enter your question, your first name, and a valid email address.

The formula used to find the height is: h = (2 * K) / c, where K is the area of the triangle.

The formula used to find the radius is: R = (a*b*c) / (4*K), where K is the area of the triangle.

The formula used to find the radius is: r = sqrt[((s-a)*(s-b)*(s-c))/s], where s is the semi-perimeter of the triangle.

The formula used is: D = 2(Bx*Cy - By*Cx) Ux = Cy(Bx² + By²) - By(Cx² + Cy²) / D Uy = Bx(Cx² + Cy²) - Cx(Bx² + By²) / D.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the type of triangle based on the entered and computed triangle values.

The inscribed circle is the largest circle that will fit within the triangle.

The circumscribed circle is a circle that passes through all three vertices of a triangle.

The formula used is K = (base x height) / 2.

The formula used is P = a + b + c.

The formula used for finding the length of the line is: ma = (1/2)sqrt[2c² + 2b² - a²].

The formula used for finding the length of the line is: mb = (1/2)sqrt[2a² + 2c² - b²].

The formula used for finding the length of the line is: mc = (1/2)sqrt[2a² + 2b² - c²].

This is the length of the median (ma), which is the line that runs from vertex A to the mid-point of side a (the opposite side).

This is the length of the median (mb), which is the line that runs from vertex B to the mid-point of side b (the opposite side).

This is the length of the median (mc), which is the line that runs from vertex C to the mid-point of side c (the opposite side).

This is the height (h) of the triangle using side a as the base.

This is the height (h) of the triangle using side b as the base.

This is the height (h) of the triangle using side c as the base.

This is the angle of vertex A of the solved triangle.

This is the angle of vertex B of the solved triangle.

This is the angle of vertex C of the solved triangle.

To save changes to previously saved entries, simply tap the Save button.

If you gave the calculator less than "A", please help me improve the calculator by telling me what I would need to do for it to earn an "A".

Clicking the "Reset" button will restore the calculator to its default settings.

Please select and "Clear" any data records you no longer need.

Note that my expertise is in creating online calculators, not necessarily in all of the subject areas they cover.

If it's not filled in, please enter the web address of the calculator as displayed in the location field at the top of the browser window (www.free-online-calculator-use.com/____.html).

This field should already be filled in if you are using a newer web browser with javascript turned on.

A Data Record is a set of calculator entries that are stored in your web browser's Local Storage.

If you have a question about the calculator's operation, please enter your question, your first name, and a valid email address.

The formula used to find the height is: h = (2 * K) / c, where K is the area of the triangle.

The formula used to find the radius is: R = (a*b*c) / (4*K), where K is the area of the triangle.

The formula used to find the radius is: r = sqrt[((s-a)*(s-b)*(s-c))/s], where s is the semi-perimeter of the triangle.

The formula used is: D = 2(Bx*Cy - By*Cx) Ux = Cy(Bx² + By²) - By(Cx² + Cy²) / D Uy = Bx(Cx² + Cy²) - Cx(Bx² + By²) / D.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the type of triangle based on the entered and computed triangle values.

The inscribed circle is the largest circle that will fit within the triangle.

The circumscribed circle is a circle that passes through all three vertices of a triangle.

The formula used is K = (base x height) / 2.

The formula used is P = a + b + c.

The formula used for finding the length of the line is: ma = (1/2)sqrt[2c² + 2b² - a²].

The formula used for finding the length of the line is: mb = (1/2)sqrt[2a² + 2c² - b²].

The formula used for finding the length of the line is: mc = (1/2)sqrt[2a² + 2b² - c²].

This is the length of the median (ma), which is the line that runs from vertex A to the mid-point of side a (the opposite side).

This is the length of the median (mb), which is the line that runs from vertex B to the mid-point of side b (the opposite side).

This is the length of the median (mc), which is the line that runs from vertex C to the mid-point of side c (the opposite side).

This is the height (h) of the triangle using side a as the base.

This is the height (h) of the triangle using side b as the base.

This is the height (h) of the triangle using side c as the base.

This is the angle of vertex A of the solved triangle.

This is the angle of vertex B of the solved triangle.

This is the angle of vertex C of the solved triangle.

To save changes to previously saved entries, simply tap the Save button.

If you gave the calculator less than "A", please help me improve the calculator by telling me what I would need to do for it to earn an "A".

Clicking the "Reset" button will restore the calculator to its default settings.

Please select and "Clear" any data records you no longer need.

Note that my expertise is in creating online calculators, not necessarily in all of the subject areas they cover.

If it's not filled in, please enter the web address of the calculator as displayed in the location field at the top of the browser window (www.free-online-calculator-use.com/____.html).

This field should already be filled in if you are using a newer web browser with javascript turned on.

A Data Record is a set of calculator entries that are stored in your web browser's Local Storage.

If you have a question about the calculator's operation, please enter your question, your first name, and a valid email address.

The formula used to find the height is: h = (2 * K) / c, where K is the area of the triangle.

The formula used to find the radius is: R = (a*b*c) / (4*K), where K is the area of the triangle.

The formula used to find the radius is: r = sqrt[((s-a)*(s-b)*(s-c))/s], where s is the semi-perimeter of the triangle.

The formula used is: D = 2(Bx*Cy - By*Cx) Ux = Cy(Bx² + By²) - By(Cx² + Cy²) / D Uy = Bx(Cx² + Cy²) - Cx(Bx² + By²) / D.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the type of triangle based on the entered and computed triangle values.

The inscribed circle is the largest circle that will fit within the triangle.

The circumscribed circle is a circle that passes through all three vertices of a triangle.

The formula used is K = (base x height) / 2.

The formula used is P = a + b + c.

The formula used for finding the length of the line is: ma = (1/2)sqrt[2c² + 2b² - a²].

The formula used for finding the length of the line is: mb = (1/2)sqrt[2a² + 2c² - b²].

The formula used for finding the length of the line is: mc = (1/2)sqrt[2a² + 2b² - c²].

This is the length of the median (ma), which is the line that runs from vertex A to the mid-point of side a (the opposite side).

This is the length of the median (mb), which is the line that runs from vertex B to the mid-point of side b (the opposite side).

This is the length of the median (mc), which is the line that runs from vertex C to the mid-point of side c (the opposite side).

This is the height (h) of the triangle using side a as the base.

This is the height (h) of the triangle using side b as the base.

This is the height (h) of the triangle using side c as the base.

This is the angle of vertex A of the solved triangle.

This is the angle of vertex B of the solved triangle.

This is the angle of vertex C of the solved triangle.

To save changes to previously saved entries, simply tap the Save button.

If you gave the calculator less than "A", please help me improve the calculator by telling me what I would need to do for it to earn an "A".

Clicking the "Reset" button will restore the calculator to its default settings.

Please select and "Clear" any data records you no longer need.

Note that my expertise is in creating online calculators, not necessarily in all of the subject areas they cover.

If it's not filled in, please enter the web address of the calculator as displayed in the location field at the top of the browser window (www.free-online-calculator-use.com/____.html).

This field should already be filled in if you are using a newer web browser with javascript turned on.

A Data Record is a set of calculator entries that are stored in your web browser's Local Storage.

If you have a question about the calculator's operation, please enter your question, your first name, and a valid email address.

The formula used to find the height is: h = (2 * K) / c, where K is the area of the triangle.

The formula used to find the radius is: R = (a*b*c) / (4*K), where K is the area of the triangle.

The formula used to find the radius is: r = sqrt[((s-a)*(s-b)*(s-c))/s], where s is the semi-perimeter of the triangle.

The formula used is: D = 2(Bx*Cy - By*Cx) Ux = Cy(Bx² + By²) - By(Cx² + Cy²) / D Uy = Bx(Cx² + Cy²) - Cx(Bx² + By²) / D.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the type of triangle based on the entered and computed triangle values.

The inscribed circle is the largest circle that will fit within the triangle.

The circumscribed circle is a circle that passes through all three vertices of a triangle.

The formula used is K = (base x height) / 2.

The formula used is P = a + b + c.

The formula used for finding the length of the line is: ma = (1/2)sqrt[2c² + 2b² - a²].

The formula used for finding the length of the line is: mb = (1/2)sqrt[2a² + 2c² - b²].

The formula used for finding the length of the line is: mc = (1/2)sqrt[2a² + 2b² - c²].

This is the length of the median (ma), which is the line that runs from vertex A to the mid-point of side a (the opposite side).

This is the length of the median (mb), which is the line that runs from vertex B to the mid-point of side b (the opposite side).

This is the length of the median (mc), which is the line that runs from vertex C to the mid-point of side c (the opposite side).

This is the height (h) of the triangle using side a as the base.

This is the height (h) of the triangle using side b as the base.

This is the height (h) of the triangle using side c as the base.

This is the angle of vertex A of the solved triangle.

This is the angle of vertex B of the solved triangle.

This is the angle of vertex C of the solved triangle.

To save changes to previously saved entries, simply tap the Save button.

If you gave the calculator less than "A", please help me improve the calculator by telling me what I would need to do for it to earn an "A".

Clicking the "Reset" button will restore the calculator to its default settings.

Please select and "Clear" any data records you no longer need.

Note that my expertise is in creating online calculators, not necessarily in all of the subject areas they cover.

If it's not filled in, please enter the web address of the calculator as displayed in the location field at the top of the browser window (www.free-online-calculator-use.com/____.html).

This field should already be filled in if you are using a newer web browser with javascript turned on.

A Data Record is a set of calculator entries that are stored in your web browser's Local Storage.

If you have a question about the calculator's operation, please enter your question, your first name, and a valid email address.

The formula used to find the height is: h = (2 * K) / c, where K is the area of the triangle.

The formula used to find the radius is: R = (a*b*c) / (4*K), where K is the area of the triangle.

The formula used to find the radius is: r = sqrt[((s-a)*(s-b)*(s-c))/s], where s is the semi-perimeter of the triangle.

The formula used is: D = 2(Bx*Cy - By*Cx) Ux = Cy(Bx² + By²) - By(Cx² + Cy²) / D Uy = Bx(Cx² + Cy²) - Cx(Bx² + By²) / D.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the type of triangle based on the entered and computed triangle values.

The inscribed circle is the largest circle that will fit within the triangle.

The circumscribed circle is a circle that passes through all three vertices of a triangle.

The formula used is K = (base x height) / 2.

The formula used is P = a + b + c.

The formula used for finding the length of the line is: ma = (1/2)sqrt[2c² + 2b² - a²].

The formula used for finding the length of the line is: mb = (1/2)sqrt[2a² + 2c² - b²].

The formula used for finding the length of the line is: mc = (1/2)sqrt[2a² + 2b² - c²].

This is the length of the median (ma), which is the line that runs from vertex A to the mid-point of side a (the opposite side).

This is the length of the median (mb), which is the line that runs from vertex B to the mid-point of side b (the opposite side).

This is the length of the median (mc), which is the line that runs from vertex C to the mid-point of side c (the opposite side).

This is the height (h) of the triangle using side a as the base.

This is the height (h) of the triangle using side b as the base.

This is the height (h) of the triangle using side c as the base.

This is the angle of vertex A of the solved triangle.

This is the angle of vertex B of the solved triangle.

This is the angle of vertex C of the solved triangle.

To save changes to previously saved entries, simply tap the Save button.

If you gave the calculator less than "A", please help me improve the calculator by telling me what I would need to do for it to earn an "A".

Clicking the "Reset" button will restore the calculator to its default settings.

Please select and "Clear" any data records you no longer need.

Note that my expertise is in creating online calculators, not necessarily in all of the subject areas they cover.

If it's not filled in, please enter the web address of the calculator as displayed in the location field at the top of the browser window (www.free-online-calculator-use.com/____.html).

This field should already be filled in if you are using a newer web browser with javascript turned on.

A Data Record is a set of calculator entries that are stored in your web browser's Local Storage.

If you have a question about the calculator's operation, please enter your question, your first name, and a valid email address.

The formula used to find the height is: h = (2 * K) / c, where K is the area of the triangle.

The formula used to find the radius is: R = (a*b*c) / (4*K), where K is the area of the triangle.

The formula used to find the radius is: r = sqrt[((s-a)*(s-b)*(s-c))/s], where s is the semi-perimeter of the triangle.

The formula used is: D = 2(Bx*Cy - By*Cx) Ux = Cy(Bx² + By²) - By(Cx² + Cy²) / D Uy = Bx(Cx² + Cy²) - Cx(Bx² + By²) / D.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the type of triangle based on the entered and computed triangle values.

The inscribed circle is the largest circle that will fit within the triangle.

The circumscribed circle is a circle that passes through all three vertices of a triangle.

The formula used is K = (base x height) / 2.

The formula used is P = a + b + c.

The formula used for finding the length of the line is: ma = (1/2)sqrt[2c² + 2b² - a²].

The formula used for finding the length of the line is: mb = (1/2)sqrt[2a² + 2c² - b²].

The formula used for finding the length of the line is: mc = (1/2)sqrt[2a² + 2b² - c²].

This is the length of the median (ma), which is the line that runs from vertex A to the mid-point of side a (the opposite side).

This is the length of the median (mb), which is the line that runs from vertex B to the mid-point of side b (the opposite side).

This is the length of the median (mc), which is the line that runs from vertex C to the mid-point of side c (the opposite side).

This is the height (h) of the triangle using side a as the base.

This is the height (h) of the triangle using side b as the base.

This is the height (h) of the triangle using side c as the base.

This is the angle of vertex A of the solved triangle.

This is the angle of vertex B of the solved triangle.

This is the angle of vertex C of the solved triangle.

To save changes to previously saved entries, simply tap the Save button.

If you gave the calculator less than "A", please help me improve the calculator by telling me what I would need to do for it to earn an "A".

Clicking the "Reset" button will restore the calculator to its default settings.

Please select and "Clear" any data records you no longer need.

Note that my expertise is in creating online calculators, not necessarily in all of the subject areas they cover.

If it's not filled in, please enter the web address of the calculator as displayed in the location field at the top of the browser window (www.free-online-calculator-use.com/____.html).

This field should already be filled in if you are using a newer web browser with javascript turned on.

A Data Record is a set of calculator entries that are stored in your web browser's Local Storage.

If you have a question about the calculator's operation, please enter your question, your first name, and a valid email address.

The formula used to find the height is: h = (2 * K) / c, where K is the area of the triangle.

The formula used to find the radius is: R = (a*b*c) / (4*K), where K is the area of the triangle.

The formula used to find the radius is: r = sqrt[((s-a)*(s-b)*(s-c))/s], where s is the semi-perimeter of the triangle.

The formula used is: D = 2(Bx*Cy - By*Cx) Ux = Cy(Bx² + By²) - By(Cx² + Cy²) / D Uy = Bx(Cx² + Cy²) - Cx(Bx² + By²) / D.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the type of triangle based on the entered and computed triangle values.

The inscribed circle is the largest circle that will fit within the triangle.

The circumscribed circle is a circle that passes through all three vertices of a triangle.

The formula used is K = (base x height) / 2.

The formula used is P = a + b + c.

The formula used for finding the length of the line is: ma = (1/2)sqrt[2c² + 2b² - a²].

The formula used for finding the length of the line is: mb = (1/2)sqrt[2a² + 2c² - b²].

The formula used for finding the length of the line is: mc = (1/2)sqrt[2a² + 2b² - c²].

This is the length of the median (ma), which is the line that runs from vertex A to the mid-point of side a (the opposite side).

This is the length of the median (mb), which is the line that runs from vertex B to the mid-point of side b (the opposite side).

This is the length of the median (mc), which is the line that runs from vertex C to the mid-point of side c (the opposite side).

This is the height (h) of the triangle using side a as the base.

This is the height (h) of the triangle using side b as the base.

This is the height (h) of the triangle using side c as the base.

This is the angle of vertex A of the solved triangle.

This is the angle of vertex B of the solved triangle.

This is the angle of vertex C of the solved triangle.

To save changes to previously saved entries, simply tap the Save button.

If you gave the calculator less than "A", please help me improve the calculator by telling me what I would need to do for it to earn an "A".

Clicking the "Reset" button will restore the calculator to its default settings.

Please select and "Clear" any data records you no longer need.

Note that my expertise is in creating online calculators, not necessarily in all of the subject areas they cover.

If it's not filled in, please enter the web address of the calculator as displayed in the location field at the top of the browser window (www.free-online-calculator-use.com/____.html).

This field should already be filled in if you are using a newer web browser with javascript turned on.

A Data Record is a set of calculator entries that are stored in your web browser's Local Storage.

If you have a question about the calculator's operation, please enter your question, your first name, and a valid email address.

The formula used to find the height is: h = (2 * K) / c, where K is the area of the triangle.

The formula used to find the radius is: R = (a*b*c) / (4*K), where K is the area of the triangle.

The formula used to find the radius is: r = sqrt[((s-a)*(s-b)*(s-c))/s], where s is the semi-perimeter of the triangle.

The formula used is: D = 2(Bx*Cy - By*Cx) Ux = Cy(Bx² + By²) - By(Cx² + Cy²) / D Uy = Bx(Cx² + Cy²) - Cx(Bx² + By²) / D.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the type of triangle based on the entered and computed triangle values.

The inscribed circle is the largest circle that will fit within the triangle.

The circumscribed circle is a circle that passes through all three vertices of a triangle.

The formula used is K = (base x height) / 2.

The formula used is P = a + b + c.

The formula used for finding the length of the line is: ma = (1/2)sqrt[2c² + 2b² - a²].

The formula used for finding the length of the line is: mb = (1/2)sqrt[2a² + 2c² - b²].

The formula used for finding the length of the line is: mc = (1/2)sqrt[2a² + 2b² - c²].

This is the length of the median (ma), which is the line that runs from vertex A to the mid-point of side a (the opposite side).

This is the length of the median (mb), which is the line that runs from vertex B to the mid-point of side b (the opposite side).

This is the length of the median (mc), which is the line that runs from vertex C to the mid-point of side c (the opposite side).

This is the height (h) of the triangle using side a as the base.

This is the height (h) of the triangle using side b as the base.

This is the height (h) of the triangle using side c as the base.

This is the angle of vertex A of the solved triangle.

This is the angle of vertex B of the solved triangle.

This is the angle of vertex C of the solved triangle.

To save changes to previously saved entries, simply tap the Save button.

If you gave the calculator less than "A", please help me improve the calculator by telling me what I would need to do for it to earn an "A".

Clicking the "Reset" button will restore the calculator to its default settings.

Please select and "Clear" any data records you no longer need.

Note that my expertise is in creating online calculators, not necessarily in all of the subject areas they cover.

If it's not filled in, please enter the web address of the calculator as displayed in the location field at the top of the browser window (www.free-online-calculator-use.com/____.html).

This field should already be filled in if you are using a newer web browser with javascript turned on.

A Data Record is a set of calculator entries that are stored in your web browser's Local Storage.

If you have a question about the calculator's operation, please enter your question, your first name, and a valid email address.

The formula used to find the height is: h = (2 * K) / c, where K is the area of the triangle.

The formula used to find the radius is: R = (a*b*c) / (4*K), where K is the area of the triangle.

The formula used to find the radius is: r = sqrt[((s-a)*(s-b)*(s-c))/s], where s is the semi-perimeter of the triangle.

The formula used is: D = 2(Bx*Cy - By*Cx) Ux = Cy(Bx² + By²) - By(Cx² + Cy²) / D Uy = Bx(Cx² + Cy²) - Cx(Bx² + By²) / D.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the type of triangle based on the entered and computed triangle values.

The inscribed circle is the largest circle that will fit within the triangle.

The circumscribed circle is a circle that passes through all three vertices of a triangle.

The formula used is K = (base x height) / 2.

The formula used is P = a + b + c.

The formula used for finding the length of the line is: ma = (1/2)sqrt[2c² + 2b² - a²].

The formula used for finding the length of the line is: mb = (1/2)sqrt[2a² + 2c² - b²].

The formula used for finding the length of the line is: mc = (1/2)sqrt[2a² + 2b² - c²].

This is the length of the median (ma), which is the line that runs from vertex A to the mid-point of side a (the opposite side).

This is the length of the median (mb), which is the line that runs from vertex B to the mid-point of side b (the opposite side).

This is the length of the median (mc), which is the line that runs from vertex C to the mid-point of side c (the opposite side).

This is the height (h) of the triangle using side a as the base.

This is the height (h) of the triangle using side b as the base.

This is the height (h) of the triangle using side c as the base.

This is the angle of vertex A of the solved triangle.

This is the angle of vertex B of the solved triangle.

This is the angle of vertex C of the solved triangle.

To save changes to previously saved entries, simply tap the Save button.

If you gave the calculator less than "A", please help me improve the calculator by telling me what I would need to do for it to earn an "A".

Clicking the "Reset" button will restore the calculator to its default settings.

Please select and "Clear" any data records you no longer need.

Note that my expertise is in creating online calculators, not necessarily in all of the subject areas they cover.

If it's not filled in, please enter the web address of the calculator as displayed in the location field at the top of the browser window (www.free-online-calculator-use.com/____.html).

This field should already be filled in if you are using a newer web browser with javascript turned on.

A Data Record is a set of calculator entries that are stored in your web browser's Local Storage.

If you have a question about the calculator's operation, please enter your question, your first name, and a valid email address.

The formula used to find the height is: h = (2 * K) / c, where K is the area of the triangle.

The formula used to find the radius is: R = (a*b*c) / (4*K), where K is the area of the triangle.

The formula used to find the radius is: r = sqrt[((s-a)*(s-b)*(s-c))/s], where s is the semi-perimeter of the triangle.

The formula used is: D = 2(Bx*Cy - By*Cx) Ux = Cy(Bx² + By²) - By(Cx² + Cy²) / D Uy = Bx(Cx² + Cy²) - Cx(Bx² + By²) / D.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the type of triangle based on the entered and computed triangle values.

The inscribed circle is the largest circle that will fit within the triangle.

The circumscribed circle is a circle that passes through all three vertices of a triangle.

The formula used is K = (base x height) / 2.

The formula used is P = a + b + c.

The formula used for finding the length of the line is: ma = (1/2)sqrt[2c² + 2b² - a²].

The formula used for finding the length of the line is: mb = (1/2)sqrt[2a² + 2c² - b²].

The formula used for finding the length of the line is: mc = (1/2)sqrt[2a² + 2b² - c²].

This is the length of the median (ma), which is the line that runs from vertex A to the mid-point of side a (the opposite side).

This is the length of the median (mb), which is the line that runs from vertex B to the mid-point of side b (the opposite side).

This is the length of the median (mc), which is the line that runs from vertex C to the mid-point of side c (the opposite side).

This is the height (h) of the triangle using side a as the base.

This is the height (h) of the triangle using side b as the base.

This is the height (h) of the triangle using side c as the base.

This is the angle of vertex A of the solved triangle.

This is the angle of vertex B of the solved triangle.

This is the angle of vertex C of the solved triangle.

To save changes to previously saved entries, simply tap the Save button.

If you gave the calculator less than "A", please help me improve the calculator by telling me what I would need to do for it to earn an "A".

Clicking the "Reset" button will restore the calculator to its default settings.

Please select and "Clear" any data records you no longer need.

Note that my expertise is in creating online calculators, not necessarily in all of the subject areas they cover.

If it's not filled in, please enter the web address of the calculator as displayed in the location field at the top of the browser window (www.free-online-calculator-use.com/____.html).

This field should already be filled in if you are using a newer web browser with javascript turned on.

A Data Record is a set of calculator entries that are stored in your web browser's Local Storage.

If you have a question about the calculator's operation, please enter your question, your first name, and a valid email address.

The formula used to find the height is: h = (2 * K) / c, where K is the area of the triangle.

The formula used to find the radius is: R = (a*b*c) / (4*K), where K is the area of the triangle.

The formula used to find the radius is: r = sqrt[((s-a)*(s-b)*(s-c))/s], where s is the semi-perimeter of the triangle.

The formula used is: D = 2(Bx*Cy - By*Cx) Ux = Cy(Bx² + By²) - By(Cx² + Cy²) / D Uy = Bx(Cx² + Cy²) - Cx(Bx² + By²) / D.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the type of triangle based on the entered and computed triangle values.

The inscribed circle is the largest circle that will fit within the triangle.

The circumscribed circle is a circle that passes through all three vertices of a triangle.

The formula used is K = (base x height) / 2.

The formula used is P = a + b + c.

The formula used for finding the length of the line is: ma = (1/2)sqrt[2c² + 2b² - a²].

The formula used for finding the length of the line is: mb = (1/2)sqrt[2a² + 2c² - b²].

The formula used for finding the length of the line is: mc = (1/2)sqrt[2a² + 2b² - c²].

This is the length of the median (ma), which is the line that runs from vertex A to the mid-point of side a (the opposite side).

This is the length of the median (mb), which is the line that runs from vertex B to the mid-point of side b (the opposite side).

This is the length of the median (mc), which is the line that runs from vertex C to the mid-point of side c (the opposite side).

This is the height (h) of the triangle using side a as the base.

This is the height (h) of the triangle using side b as the base.

This is the height (h) of the triangle using side c as the base.

This is the angle of vertex A of the solved triangle.

This is the angle of vertex B of the solved triangle.

This is the angle of vertex C of the solved triangle.

To save changes to previously saved entries, simply tap the Save button.

If you gave the calculator less than "A", please help me improve the calculator by telling me what I would need to do for it to earn an "A".

Clicking the "Reset" button will restore the calculator to its default settings.

Please select and "Clear" any data records you no longer need.

Note that my expertise is in creating online calculators, not necessarily in all of the subject areas they cover.

If it's not filled in, please enter the web address of the calculator as displayed in the location field at the top of the browser window (www.free-online-calculator-use.com/____.html).

This field should already be filled in if you are using a newer web browser with javascript turned on.

A Data Record is a set of calculator entries that are stored in your web browser's Local Storage.

If you have a question about the calculator's operation, please enter your question, your first name, and a valid email address.

The formula used to find the height is: h = (2 * K) / c, where K is the area of the triangle.

The formula used to find the radius is: R = (a*b*c) / (4*K), where K is the area of the triangle.

The formula used to find the radius is: r = sqrt[((s-a)*(s-b)*(s-c))/s], where s is the semi-perimeter of the triangle.

The formula used is: D = 2(Bx*Cy - By*Cx) Ux = Cy(Bx² + By²) - By(Cx² + Cy²) / D Uy = Bx(Cx² + Cy²) - Cx(Bx² + By²) / D.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the type of triangle based on the entered and computed triangle values.

The inscribed circle is the largest circle that will fit within the triangle.

The circumscribed circle is a circle that passes through all three vertices of a triangle.

The formula used is K = (base x height) / 2.

The formula used is P = a + b + c.

The formula used for finding the length of the line is: ma = (1/2)sqrt[2c² + 2b² - a²].

The formula used for finding the length of the line is: mb = (1/2)sqrt[2a² + 2c² - b²].

The formula used for finding the length of the line is: mc = (1/2)sqrt[2a² + 2b² - c²].

This is the length of the median (ma), which is the line that runs from vertex A to the mid-point of side a (the opposite side).

This is the length of the median (mb), which is the line that runs from vertex B to the mid-point of side b (the opposite side).

This is the length of the median (mc), which is the line that runs from vertex C to the mid-point of side c (the opposite side).

This is the height (h) of the triangle using side a as the base.

This is the height (h) of the triangle using side b as the base.

This is the height (h) of the triangle using side c as the base.

This is the angle of vertex A of the solved triangle.

This is the angle of vertex B of the solved triangle.

This is the angle of vertex C of the solved triangle.

To save changes to previously saved entries, simply tap the Save button.

If you gave the calculator less than "A", please help me improve the calculator by telling me what I would need to do for it to earn an "A".

Clicking the "Reset" button will restore the calculator to its default settings.

Please select and "Clear" any data records you no longer need.

Note that my expertise is in creating online calculators, not necessarily in all of the subject areas they cover.

If it's not filled in, please enter the web address of the calculator as displayed in the location field at the top of the browser window (www.free-online-calculator-use.com/____.html).

This field should already be filled in if you are using a newer web browser with javascript turned on.

A Data Record is a set of calculator entries that are stored in your web browser's Local Storage.

If you have a question about the calculator's operation, please enter your question, your first name, and a valid email address.

The formula used to find the height is: h = (2 * K) / c, where K is the area of the triangle.

The formula used to find the radius is: R = (a*b*c) / (4*K), where K is the area of the triangle.

The formula used to find the radius is: r = sqrt[((s-a)*(s-b)*(s-c))/s], where s is the semi-perimeter of the triangle.

The formula used is: D = 2(Bx*Cy - By*Cx) Ux = Cy(Bx² + By²) - By(Cx² + Cy²) / D Uy = Bx(Cx² + Cy²) - Cx(Bx² + By²) / D.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the type of triangle based on the entered and computed triangle values.

The inscribed circle is the largest circle that will fit within the triangle.

The circumscribed circle is a circle that passes through all three vertices of a triangle.

The formula used is K = (base x height) / 2.

The formula used is P = a + b + c.

The formula used for finding the length of the line is: ma = (1/2)sqrt[2c² + 2b² - a²].

The formula used for finding the length of the line is: mb = (1/2)sqrt[2a² + 2c² - b²].

The formula used for finding the length of the line is: mc = (1/2)sqrt[2a² + 2b² - c²].

This is the length of the median (ma), which is the line that runs from vertex A to the mid-point of side a (the opposite side).

This is the length of the median (mb), which is the line that runs from vertex B to the mid-point of side b (the opposite side).

This is the length of the median (mc), which is the line that runs from vertex C to the mid-point of side c (the opposite side).

This is the height (h) of the triangle using side a as the base.

This is the height (h) of the triangle using side b as the base.

This is the height (h) of the triangle using side c as the base.

This is the angle of vertex A of the solved triangle.

This is the angle of vertex B of the solved triangle.

This is the angle of vertex C of the solved triangle.

To save changes to previously saved entries, simply tap the Save button.

If you gave the calculator less than "A", please help me improve the calculator by telling me what I would need to do for it to earn an "A".

Clicking the "Reset" button will restore the calculator to its default settings.

Please select and "Clear" any data records you no longer need.

Note that my expertise is in creating online calculators, not necessarily in all of the subject areas they cover.

If it's not filled in, please enter the web address of the calculator as displayed in the location field at the top of the browser window (www.free-online-calculator-use.com/____.html).

This field should already be filled in if you are using a newer web browser with javascript turned on.

A Data Record is a set of calculator entries that are stored in your web browser's Local Storage.

If you have a question about the calculator's operation, please enter your question, your first name, and a valid email address.

The formula used to find the height is: h = (2 * K) / c, where K is the area of the triangle.

The formula used to find the radius is: R = (a*b*c) / (4*K), where K is the area of the triangle.

The formula used to find the radius is: r = sqrt[((s-a)*(s-b)*(s-c))/s], where s is the semi-perimeter of the triangle.

The formula used is: D = 2(Bx*Cy - By*Cx) Ux = Cy(Bx² + By²) - By(Cx² + Cy²) / D Uy = Bx(Cx² + Cy²) - Cx(Bx² + By²) / D.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the type of triangle based on the entered and computed triangle values.

The inscribed circle is the largest circle that will fit within the triangle.

The circumscribed circle is a circle that passes through all three vertices of a triangle.

The formula used is K = (base x height) / 2.

The formula used is P = a + b + c.

The formula used for finding the length of the line is: ma = (1/2)sqrt[2c² + 2b² - a²].

The formula used for finding the length of the line is: mb = (1/2)sqrt[2a² + 2c² - b²].

The formula used for finding the length of the line is: mc = (1/2)sqrt[2a² + 2b² - c²].

This is the length of the median (ma), which is the line that runs from vertex A to the mid-point of side a (the opposite side).

This is the length of the median (mb), which is the line that runs from vertex B to the mid-point of side b (the opposite side).

This is the length of the median (mc), which is the line that runs from vertex C to the mid-point of side c (the opposite side).

This is the height (h) of the triangle using side a as the base.

This is the height (h) of the triangle using side b as the base.

This is the height (h) of the triangle using side c as the base.

This is the angle of vertex A of the solved triangle.

This is the angle of vertex B of the solved triangle.

This is the angle of vertex C of the solved triangle.

To save changes to previously saved entries, simply tap the Save button.

If you gave the calculator less than "A", please help me improve the calculator by telling me what I would need to do for it to earn an "A".

Clicking the "Reset" button will restore the calculator to its default settings.

Please select and "Clear" any data records you no longer need.

Note that my expertise is in creating online calculators, not necessarily in all of the subject areas they cover.

If it's not filled in, please enter the web address of the calculator as displayed in the location field at the top of the browser window (www.free-online-calculator-use.com/____.html).

This field should already be filled in if you are using a newer web browser with javascript turned on.

A Data Record is a set of calculator entries that are stored in your web browser's Local Storage.

If you have a question about the calculator's operation, please enter your question, your first name, and a valid email address.

The formula used to find the height is: h = (2 * K) / c, where K is the area of the triangle.

The formula used to find the radius is: R = (a*b*c) / (4*K), where K is the area of the triangle.

The formula used to find the radius is: r = sqrt[((s-a)*(s-b)*(s-c))/s], where s is the semi-perimeter of the triangle.

The formula used is: D = 2(Bx*Cy - By*Cx) Ux = Cy(Bx² + By²) - By(Cx² + Cy²) / D Uy = Bx(Cx² + Cy²) - Cx(Bx² + By²) / D.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the type of triangle based on the entered and computed triangle values.

The inscribed circle is the largest circle that will fit within the triangle.

The circumscribed circle is a circle that passes through all three vertices of a triangle.

The formula used is K = (base x height) / 2.

The formula used is P = a + b + c.

The formula used for finding the length of the line is: ma = (1/2)sqrt[2c² + 2b² - a²].

The formula used for finding the length of the line is: mb = (1/2)sqrt[2a² + 2c² - b²].

The formula used for finding the length of the line is: mc = (1/2)sqrt[2a² + 2b² - c²].

This is the length of the median (ma), which is the line that runs from vertex A to the mid-point of side a (the opposite side).

This is the length of the median (mb), which is the line that runs from vertex B to the mid-point of side b (the opposite side).

This is the length of the median (mc), which is the line that runs from vertex C to the mid-point of side c (the opposite side).

This is the height (h) of the triangle using side a as the base.

This is the height (h) of the triangle using side b as the base.

This is the height (h) of the triangle using side c as the base.

This is the angle of vertex A of the solved triangle.

This is the angle of vertex B of the solved triangle.

This is the angle of vertex C of the solved triangle.

To save changes to previously saved entries, simply tap the Save button.

If you gave the calculator less than "A", please help me improve the calculator by telling me what I would need to do for it to earn an "A".

Clicking the "Reset" button will restore the calculator to its default settings.

Please select and "Clear" any data records you no longer need.

Note that my expertise is in creating online calculators, not necessarily in all of the subject areas they cover.

If it's not filled in, please enter the web address of the calculator as displayed in the location field at the top of the browser window (www.free-online-calculator-use.com/____.html).

This field should already be filled in if you are using a newer web browser with javascript turned on.

A Data Record is a set of calculator entries that are stored in your web browser's Local Storage.

If you have a question about the calculator's operation, please enter your question, your first name, and a valid email address.

The formula used to find the height is: h = (2 * K) / c, where K is the area of the triangle.

The formula used to find the radius is: R = (a*b*c) / (4*K), where K is the area of the triangle.

The formula used to find the radius is: r = sqrt[((s-a)*(s-b)*(s-c))/s], where s is the semi-perimeter of the triangle.

The formula used is: D = 2(Bx*Cy - By*Cx) Ux = Cy(Bx² + By²) - By(Cx² + Cy²) / D Uy = Bx(Cx² + Cy²) - Cx(Bx² + By²) / D.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the type of triangle based on the entered and computed triangle values.

The inscribed circle is the largest circle that will fit within the triangle.

The circumscribed circle is a circle that passes through all three vertices of a triangle.

The formula used is K = (base x height) / 2.

The formula used is P = a + b + c.

The formula used for finding the length of the line is: ma = (1/2)sqrt[2c² + 2b² - a²].

The formula used for finding the length of the line is: mb = (1/2)sqrt[2a² + 2c² - b²].

The formula used for finding the length of the line is: mc = (1/2)sqrt[2a² + 2b² - c²].

This is the length of the median (ma), which is the line that runs from vertex A to the mid-point of side a (the opposite side).

This is the length of the median (mb), which is the line that runs from vertex B to the mid-point of side b (the opposite side).

This is the length of the median (mc), which is the line that runs from vertex C to the mid-point of side c (the opposite side).

This is the height (h) of the triangle using side a as the base.

This is the height (h) of the triangle using side b as the base.

This is the height (h) of the triangle using side c as the base.

This is the angle of vertex A of the solved triangle.

This is the angle of vertex B of the solved triangle.

This is the angle of vertex C of the solved triangle.

To save changes to previously saved entries, simply tap the Save button.

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The formula used to find the height is: h = (2 * K) / c, where K is the area of the triangle.

The formula used to find the radius is: R = (a*b*c) / (4*K), where K is the area of the triangle.

The formula used to find the radius is: r = sqrt[((s-a)*(s-b)*(s-c))/s], where s is the semi-perimeter of the triangle.

The formula used is: D = 2(Bx*Cy - By*Cx) Ux = Cy(Bx² + By²) - By(Cx² + Cy²) / D Uy = Bx(Cx² + Cy²) - Cx(Bx² + By²) / D.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the circumcenter coordinate based on the three vertex coordinates.

This is the type of triangle based on the entered and computed triangle values.

The inscribed circle is the largest circle that will fit within the triangle.

The circumscribed circle is a circle that passes through all three vertices of a triangle.

The formula used is K = (base x height) / 2.

The formula used is P = a + b + c.

The formula used for finding the length of the line is: ma = (1/2)sqrt[2c² + 2b² - a²].

The formula used for finding the length of the line is: mb = (1/2)sqrt[2a² + 2c² - b²].

The formula used for finding the length of the line is: mc = (1/2)sqrt[2a² + 2b² - c²].

This is the length of the median (ma), which is the line that runs from vertex A to the mid


The GRE Quantitative Reasoning section assesses a test taker's basic mathematical skills, understanding of elementary mathematical concepts, and the ability to reason quantitatively and to model and solve problems with quantitative methods. The section has four content areas:

1. Arithmetic: This includes properties and types of integers, arithmetic operations, exponents and roots, estimation, percent, ratio, rate, absolute value, the number line, decimal representation, and sequences of numbers.
2. Algebra: This includes operations with exponents, factoring and simplifying algebraic expressions, relations, functions, equations and inequalities, solving linear and quadratic equations and inequalities, and coordinate geometry.
3. Geometry: This includes parallel and perpendicular lines, circles, triangles, quadrilaterals, other polygons, congruent and similar figures, three-dimensional figures, area, perimeter, volume, and the Pythagorean theorem.
4. Data analysis: This includes basic descriptive statistics, interpretation of data in tables and graphs, elementary probability, conditional probability, random variables, and probability distributions.

The GRE Quantitative Reasoning section has four question types:
1. Quantitative Comparison: This type of question asks a test taker to compare two quantities and determine which of the statements describes the comparison.
2. Multiple Choice – Select One Answer: This type of question is similar to the traditional multiple choice of selecting one answer from a list of choices.
3. Multiple Choice – Select One or More Answer: This type of question asks a test taker to select one or more from a list of choices.
4. Numeric Entry: This type of question asks examinees to either input their answer as an integer or a decimal in a single answer box, or to enter it as a fraction in two separate boxes.

The Data Interpretation questions ask a test taker to interpret or analyze data grouped together, in a table, graph, or other data presentation. These questions may be Multiple Choice or Numeric Entry.

The GRE Quantitative Reasoning has two sections with 20 questions each, and each section may be taken for 35 minutes. The Quantitative Reasoning Score is based on the test taker's number of correct responses to the questions, and it will also reflect the difficulty of each level section. The score may range from 130 to 170 in 1-point increments, with an average score of 152.57.

A good GRE Quantitative Reasoning score is 160, which falls in the 75th percentile. A score of 166 is an excellent one, which puts an examiner on the 90th percentile. A score of below 154 is a bad score, which falls in the 50th percentile. A score of 147 is worse, which puts an examiner on the 25th percentile.

GRE preparation courses can help with Quantitative Reasoning by providing structured content, practice problems, and test-taking strategies tailored to the Quant section. These courses can significantly boost one's performance in Quantitative Reasoning.

Sample questions for each question type are available, including Quantitative Comparison, Multiple Choice – Select One Answer, Multiple Choice – Select One or More Answer, and Numeric Entry. Data Interpretation sample questions are also available, which ask test takers to interpret or analyze data grouped together, in a table, graph, or other data presentation.

References for sample questions and more information are available at the official ETS website.


The Gemara describes a process where 24 lots were divided, and each of the four mishmaros (divisions) received six lots. To calculate the probability of this division, we can use the combination formula. Initially, it seemed that the first person, Yedaya, picked six names, including his own, from the 24 lots. Using the combination formula, the number of possible combinations for Yedaya's pick would be 24C6, which is calculated as 24! / (24-6)! x 6!. This can be simplified to 24 x 23 x 22 x 21 x 20 x 19 / 6 x 5 x 4 x 3 x 2 x 1.

However, according to Rashi's explanation in Arachin 12, the process was slightly different. The head of each mishmar picked his own name first, and then the other five names. This changes the calculation to 24 x 18 x 23C5 x 12 x 17C5 / 11C5.

Another perspective is offered by Rashash, who suggests that there were four separate raffles of six lots each, which determined the order of the subdivisions within each mishmar. This explanation simplifies the calculation and implies that there was no miraculous intervention in the division of the lots.

The probability of this division can be expressed as 1 / (24C6 x 18C6 x 12C6), which equals 1 / (134596 x 18564 x 924). However, taking into account Rashi's explanation, the probability would be 1 / (24 x 18 x 23C5 x 12 x 17C5 / 11C5).

It's worth noting that the exact process and calculation may be complex and require further investigation of the commentators' explanations.


Ronald Fagin's characterization of NP is based on logic, without references to Turing machines or polynomial time. In his 1974 paper, "Generalized first-order spectra and polynomial-time recognizable sets," Fagin shows that NP consists of exactly the languages expressible with existential second-order formulas. For example, consider a graph G described by an edge relation E(i,j), and we can define whether G is k-colorable by using existential second-order formulas.

Every language in NP has an existential second-order characterization with binary predicates and a universal first-order part. Stockmeyer generalizes Fagin's result to characterize the polynomial-time hierarchy with second-order formulas. Fagin's result started the area of descriptive complexity, which characterizes many common complexity classes in various logics and has connections to the complexity of database queries.

An existential second-order formula about graphs is a formula of the form ∃ Q_{1} ... ∃ Q_{k} S( E, Q_{1}, ..., Q_{k}), where E represents the edge relation, Q_{1}, ..., Q_{k} are existentially quantified predicates of arbitrary arity, and S(E, Q_{1}, ..., Q_{k}) is a first-order formula. For instance, 3-colorability of a graph can be expressed by an existential second-order formula ∃ Q_{1} ∃ Q_{2} ∃ Q_{3} S(E, Q_{1}, Q_{2}, Q_{3}), where Q_{1}, Q_{2}, and Q_{3} are unary predicates representing the 3 colors.

Fagin's theorem states that if T is a class of graphs that is closed under isomorphism, then T is in NP if and only if T is defined by an existential second-order formula. However, it is an open problem whether we really need to allow existentially quantified predicates of arbitrary arity. In the case of graphs, it is conceivable that there are NP properties that require existentially quantified predicates of arbitrarily large arity, but it is also conceivable that we can capture every NP property of graphs by allowing only a single existentially quantified binary predicate.

For arbitrary structures, the characterization of NP in Fagin's theorem continues to hold, but it is known that existentially quantified binary predicates do not suffice. Ajtai proved that if we consider structures consisting of a single m-ary relation, then the statement "The number of m-tuples in the relation is even" cannot be captured in existential second-order logic with any number of existentially quantified predicates of arity less than m.

Fagin's result has been used to characterize the class MAX-SNP of optimization problems. The concept of probabilistically checkable proof itself originally comes from a second-order view of NP that originated from Fagin's paper. The PCP Theorem shows that MAX-SNP hard problems cannot be approximated within an arbitrary constant unless P=NP.

The class of NP properties that can be defined using only unary existentially quantified predicates is called "monadic NP." However, not every NP graph property is in monadic NP. For example, graph connectivity is not in monadic NP, although graph nonconnectivity is. This result is especially intriguing, since monadic NP is not closed under complement.

Fagin's theorem provides a logical characterization of NP, which is an important class in computational complexity theory. The theorem states that NP is equivalent to the class of languages that can be expressed using existential second-order formulas. This characterization has far-reaching implications for our understanding of NP and its relationship to other complexity classes.

In terms of the importance of NP, Fagin's theorem provides a strong argument for why NP is a fundamental class. NP is about searching for "solutions" to "problems," and the theorem shows that these solutions can be verified using existential second-order formulas. This provides a logical framework for understanding the nature of NP and its relationship to other complexity classes.

Overall, Fagin's theorem is a significant result in computational complexity theory, and its implications continue to be felt in the field. The theorem provides a logical characterization of NP, which has far-reaching implications for our understanding of the class and its relationship to other complexity classes.

Note: The original text included some comments and discussions, but they have been removed to provide a more concise and refined version of the text. The refined text maintains the original length and content, while removing irrelevant information and reorganizing the content into a logically structured and information-dense version.


You can still do well on SAT math, even if you dislike it or feel intimidated. The key is to not get intimidated, as the SAT only tests arithmetic, algebra, and geometry, without calculus, pre-calculus, or trigonometry. Within these subjects, the SAT focuses on a limited number of concepts.

In geometry, the SAT tests the three basic shapes: rectangles, triangles, and circles. For each shape, it only tests a few basic aspects. For example, in triangles, the SAT tests:

* Area: A = ½ bh
* Perimeter: P = s + s + s
* Angle measure: Rule of 180 (the three angle measures add up to 180)
* Side length in a right triangle: Pythagorean Theorem (a^2 + b^2 = c^2) or its shortcuts, such as special right triangles (30-60-90, 45-45-90) and Pythagorean triples (e.g., 3-4-5, 5-12-13)

These concepts are straightforward, but the SAT tests them in difficult ways. For instance, given the area of a triangle, you may need to work backwards to find the height.

Consider an example: An isosceles right triangle has an area of 12.5. What is the height of the triangle? To solve this, you can use the area formula: A = ½ bh. Given that the triangle is isosceles, the base and height are equal, so you can replace b with h: 12.5 = ½ h^2. Dividing both sides by ½ and taking the square root of each side gives you the height: 5 = h.

While the concepts involved may not be complex, the problems can be tricky. However, by practicing with the various algebraic and geometric formulas the SAT tests, you can work your way to correct answers. The SAT provides formulas, so it's essential to understand how to apply them in different scenarios. With practice, you can build confidence and improve your performance on SAT math.


To find two geometric means between 4 and 64, we need to determine a geometric sequence. Let's denote the first geometric mean as "a" and the second as "b".

We are given that the common ratio is "r". Therefore, we can express "a" and "b" in terms of "r":
- a = 4r
- b = ar = 4r^2
- 64 = br = 4r^3

Solving for "r" yields:
- 4r^3 = 64
- r^3 = 16
- r = 2^(4/3) = 2 * 2^(1/3) = 2 * ∛2 = 2∛2

Using this value of "r", we can calculate "a" and "b":
- a = 4r = 4 * 2∛2 = 8∛2
- b = ar = 8∛2 * 2∛2 = 16∛4

Thus, the two geometric means between 4 and 64 are a = 8∛2 and b = 16∛4.


Sorting means arranging the elements of an array in either ascending or descending order. This makes it easier to find elements quickly, as seen in Binary Search, which works on sorted arrays and can find elements much faster than Linear Search.

In programming, there are multiple ways to sort arrays. The ICSE Computer Applications syllabus prescribes two methods: Bubble Sort and Selection Sort.

## Sorting with Bubble Sort

Bubble Sort can be understood through a simple real-world example. Suppose we have four cards with numbers 9, 5, 2, and 3, and we want to arrange them in ascending order using Bubble Sort. After sorting, the order should be 2, 3, 5, and 9.

In Bubble Sort, we compare adjacent elements and swap them if they are in the wrong order. This process is repeated until the array is sorted.

### First Pass

To sort the cards using Bubble Sort, we compare the first two cards. Since 9 is greater than 5, we swap them. Then, we compare 9 with 2 and swap them because 9 is greater than 2. Next, we compare 9 with 3 and swap them because 9 is greater than 3. This completes the first pass, and the largest card, 9, is now in its correct position.

### Second Pass

In the second pass, we compare 5 and 2, and swap them because 5 is greater than 2. Then, we compare 5 and 3, and swap them because 5 is greater than 3. This completes the second pass, and the last two cards are now in their correct positions.

### Third Pass

In the third pass, we compare 2 and 3, but they are already in the correct order, so no swap is needed. This completes the third pass, and the cards are now sorted in ascending order.

## Bubble Sort Java Program

The Bubble Sort algorithm can be implemented in Java as follows:

```java
public class KboatBubbleSort {
    public static void displayArray(int arr[]) {
        for (int i = 0; i < arr.length; i++) {
            System.out.print(arr[i] + " ");
        }
        System.out.println();
    }

    public static void main(String args[]) {
        int arr[] = {9, 5, 2, 3};
        int n = arr.length;

        // Bubble Sort
        for (int i = 0; i < n - 1; i++) {
            System.out.println("\nPass Number: " + (i + 1));
            for (int j = 0; j < n - i - 1; j++) {
                System.out.println("\nComparing " + arr[j] + " and " + arr[j + 1]);
                if (arr[j] > arr[j + 1]) {
                    System.out.println(arr[j] + " is greater than " + arr[j + 1]);
                    System.out.println("Swapping " + arr[j] + " and " + arr[j + 1]);
                    int t = arr[j];
                    arr[j] = arr[j + 1];
                    arr[j + 1] = t;
                    System.out.println("Array after swap: ");
                    displayArray(arr);
                }
            }
        }
        System.out.println("\nSorted Array:");
        displayArray(arr);
    }
}
```

## Simplified Bubble Sort Java Program

The Bubble Sort program can be simplified by removing the extra print statements and the `displayArray` method:

```java
public class KboatBubbleSort {
    public static void main(String args[]) {
        int arr[] = {9, 5, 2, 3};
        int n = arr.length;

        // Bubble Sort
        for (int i = 0; i < n - 1; i++) {
            for (int j = 0; j < n - i - 1; j++) {
                if (arr[j] > arr[j + 1]) {
                    int t = arr[j];
                    arr[j] = arr[j + 1];
                    arr[j + 1] = t;
                }
            }
        }
        System.out.println("Sorted Array:");
        for (int i = 0; i < n; i++) {
            System.out.print(arr[i] + " ");
        }
    }
}
```

The Bubble Sort algorithm works by repeatedly comparing adjacent elements and swapping them if they are in the wrong order. This process is repeated until the array is sorted. The algorithm can be summarized in the following steps:

1. Compare the first two elements and swap them if they are in the wrong order.
2. Compare the second and third elements and swap them if they are in the wrong order.
3. Repeat step 2 until the end of the array is reached.
4. Repeat steps 1-3 until the array is sorted.

The time complexity of Bubble Sort is O(n^2), where n is the number of elements in the array. This makes it less efficient than other sorting algorithms like QuickSort and MergeSort for large datasets. However, Bubble Sort is simple to implement and can be useful for small datasets or educational purposes.


# Random Variables And Probability Distributions Examples

## Introduction
In probability theory, a probability density function (PDF) is a function whose value at any given sample or point in the sample space can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample.

## Probability Density Function
There are two types of random variables: discrete random variables and continuous random variables. The values of a discrete random variable are countable, which means the values are obtained by counting. The values of a continuous random variable are uncountable, which means the values are not obtained by counting, but by measuring.

## Discrete Probability Distribution Function
A discrete probability distribution function (PDF) has two characteristics:
1. Each probability is between zero and one, inclusive.
2. The sum of the probabilities is one.

## Examples
- A child psychologist is interested in the number of times a newborn baby's crying wakes its mother after midnight. For a random sample of 50 mothers, the following information was obtained: X takes on the values 0, 1, 2, 3, 4, 5.
- A hospital researcher is interested in the number of times the average post-op patient will ring the nurse during a hour shift. For a random sample of 50 patients, the following information was obtained.

## Constructing a Probability Distribution Table
Suppose Nancy has classes three days a week. She attends classes three days a week 80 percent of the time, two days 15 percent of the time, one day 4 percent of the time, and no days 1 percent of the time. Construct a probability distribution table (called a PDF table) with two columns labeled x and P(x).

## Continuous Random Variable
A continuous random variable takes on an uncountably infinite number of possible values. We use a probability density function (p.d.f.) to specify the probability of the random variable falling within a particular range of values.

## Probability Distribution
Probability Distribution is a statistical function that links or lists all the possible outcomes a random variable can take, in any random process, with its corresponding probability of occurrence. Values of a random variable change based on the underlying probability distribution.

## Conclusion
Random variables play a vital role in probability distributions and serve as the base for probability distributions. Understanding the basics of random variables and probability distributions is essential for further study in statistics and probability theory.


# Cumulative Sum of Last n Entries of Column Vectors

The problem involves calculating the cumulative sum for each category of a column vector "Name" but starting from the last n entries instead of the first entry. The given data consists of two column vectors: "Name" (300x1 Categorical) and "Amount" (300x1 Double).

## Example

The desired result is illustrated in the "CumAmount" column of the example table. For category B, the cumulative sum starts with 4, then becomes 7 (4+3), and finally becomes 4 (3+1), ignoring the first "Amount" value of B.

## Solution

To solve this problem, the following steps can be taken:

1. Sort the table to get the ordering and unique indices for each group.
2. Specify the value of n, which represents the number of last entries to consider for the cumulative sum.
3. Use the `conv` function to calculate the cumulative sum for each group, prepending zeros to the amount values to ensure the convolution works correctly.
4. Accumulate the results by group and take the convolution of each group.
5. Undo the sorting operation to match the original order.

## Code

```matlab
T = readtable('data.txt');
[Tsorted, sortidx] = sortrows(T, "C");
[~, ~, uniqueix] = unique(Tsorted.C, 'stable');
n = 2;
oneToN = (1:numel(uniqueix)).';
sumXminusN = @(x){conv([0;Tsorted.A(sort(x))],ones(n,1),'valid')};
ccac = accumarray(uniqueix, oneToN, [], sumXminusN);
ccaf = vertcat(ccac{:});
CA(sortidx, 1) = ccaf;
assert(isequal(CA,T.CA))
```

This solution works for n=2 and may require adjustments for other values of n.


To solve the equation 2.5/1.25 = 0.5/x, cross multiply to get 2.5x = 0.5 * 1.25. This simplifies to 2.5x = 0.625. Dividing both sides by 2.5 gives x = 0.625 / 2.5, which equals 0.25.

If 75 widgets are processed in 3.5 hours by 1 person, to find the time it takes for 9 people to process 1650 widgets, first determine the rate at which one person processes widgets. Then, calculate the rate for 9 people and use this to find the time required to process 1650 widgets.

In a 50-mile walk, if you walk 40% of the distance the first day and 1/3 of the remaining distance on the second day, you cover 40% of 50 miles on the first day, which is 0.4 * 50 = 20 miles. The remaining distance is 50 - 20 = 30 miles. On the second day, you walk 1/3 of 30 miles, which is (1/3) * 30 = 10 miles. The total distance covered in two days is 20 + 10 = 30 miles. Therefore, you have 50 - 30 = 20 miles left to walk.

If you walk at 3 miles per hour and jog at 6 miles per hour, and you want to cover more than 18 miles in less than 5 hours, let's denote the time spent walking as w and the time spent jogging as j. The total time spent is w + j, and this must be less than 5 hours. The total distance covered is 3w + 6j, and this must be more than 18 miles.

A woman can bicycle 66 miles in the same time it takes her to walk 24 miles. She rides 7 mph faster than she walks. If r is her walking rate, then her bicycling rate is r + 7. The time to walk 24 miles is 24/r, and the time to bicycle 66 miles is 66/(r + 7). Since these times are equal, we have 24/r = 66/(r + 7).

If Carson walks 6.5 miles in 2 hours, his rate is 6.5 / 2 = 3.25 miles per hour. To walk 16.25 miles at this rate, it will take him 16.25 / 3.25 hours.

John walked 10 miles on Saturday, with the first 5 miles at speed x and the second 5 miles at speed 2x. The time for the first part is 5/x, and the time for the second part is 5/2x. The total time is 5/x + 5/2x.

If you walk north for 2.5 miles and then east for 3 miles, your distance from the starting point can be found using the Pythagorean theorem, which gives the distance as the square root of (2.5^2 + 3^2).

Evan walks 3 miles north and then 2 miles east. The shortest distance he can walk to get back to the starting point is the hypotenuse of a right triangle with legs 3 miles and 2 miles, which is the square root of (3^2 + 2^2).

A truck travels 75 miles in 1.5 hours, so its speed is 75 / 1.5 = 50 miles per hour. To travel 325 miles at this speed, it will take 325 / 50 hours.

Joe walks 4 miles north, 9 miles east, 8 miles north, and 7 miles east. To walk straight back to the starting point, he must cover the net north distance and the net east distance. The net north distance is 4 + 8 = 12 miles, and the net east distance is 9 + 7 = 16 miles. The distance he must walk to get back is the square root of (12^2 + 16^2).


Insertion sort is an intuitive algorithm that humans use naturally when sorting cards. In this lesson, we'll cover how to implement this algorithm in TypeScript/JavaScript, why it's called insertion sort, and its complexity.

We'll create a function for insertion sort that takes a number array and returns a sorted number array. To begin, we create a copy of the original array using slice. The basic idea of insertion sort is to divide the array into a sorted left section and an unsorted right section. We move an item one by one from the unsorted right section into its rightful place in the sorted left section.

The code follows a similar pattern. We look through all the items in the array, skipping the first one, as a single item is already sorted. We select the current array item and store it in the current variable, creating a hole at the i-th position in the array. We use j to keep track of which item we are comparing against the current item, continuing comparisons until we arrive at the head of the array or the item at the j-th position is no longer smaller than the current item.

Within the loop, we slide the item at the j-th position to the right, then continue the loop to test against the next j-th index. Once the loop terminates, j is either less than zero or the item at j is smaller than the current item. In both cases, we put the current item at the (j+1)-th position.

To illustrate this, let's sort the array 4, 3, 2, 1. We start with a single sorted item of 4, then put 3 into its rightful place, followed by 2, and finally 1. The algorithm is called insertion sort because we insert the current item one by one into its rightful place in the sorted subsection of the array.

The time complexity of the algorithm can be represented by the summation 1, 2, 3, up till n-1, which is equivalent to n(n-1)/2 or O(n^2). The space complexity is O(1) since we are doing the array item move in place.

Note: The original text mentioned the space complexity as O(n), but it seems to be an error, as the algorithm only uses a constant amount of additional space to store the current item and other variables, making the space complexity O(1).


**Properties of Parallel Lines**

To solve this problem, we need to understand the properties of parallel lines, specifically:

* Alternate angles: $\angle 3 = \angle 2$
* Corresponding angles: $\angle 1 = \angle 2$
* Vertical opposite angles: $\angle 1 = \angle 3$

**Step-by-Step Solution**

Suppose we have three lines: AB, CD, and XY. We are given that AB and XY are parallel to CD. To prove that AB and XY are parallel to each other, we draw a transversal line PQ that intersects AB at point L, CD at point M, and XY at point N.

Since $AB\parallel CD$ and PQ is a transversal line, we have:

$\angle BLM = \angle DMP$ (Corresponding angles) ....................equation 1.

Similarly, since $XY\parallel CD$ and PQ is a transversal line, we have:

$\angle YNP = \angle DMP$ (Corresponding angles) ...................equation 2.

From equations 1 and 2, we can conclude that $\angle BLM = \angle YNP$.

Using the reverse statement of the property of parallel lines, which states that if corresponding angles are equal, then the lines are parallel, we can conclude that:

$AB\parallel XY$

**Alternative Method**

We can also solve this problem using an alternative method:

$AB\parallel CD$ and PQ is a transversal line.

$\therefore \angle ALM = \angle CMN$ (Corresponding angles) ....................equation 1.

$XY\parallel CD$ and PQ is a transversal line.

$\therefore \angle XNP = \angle CMN$ (Corresponding angles) ...................equation 2.

From equations 1 and 2, we can conclude that $\angle ALM = \angle XNP$.

Using the reverse statement of the property of parallel lines, we can conclude that:

$\therefore AB\parallel XY$

**Conclusion**

The given statement is true: if two lines are parallel to a third line, then they are parallel to each other.


# Fun with Complex Numbers

## Summary
Trigonometry can answer questions that algebra cannot, such as finding the square root of i and the logarithm of a negative number. Complex numbers can be plotted on a complex plane, making it easier to visualize addition and subtraction. However, for multiplication, division, and finding powers and roots, complex numbers are easier to work with in polar form.

## Polar Form
In polar form, a complex number is represented by its length (r) and direction (θ) from the origin. The length, or absolute value, is calculated using the Pythagorean theorem: r = √(a² + b²), where a and b are the real and imaginary components of the complex number. The direction, or angle, can be found using the arctangent function: θ = arctan(b/a).

For example, the complex number -3 + 5i has a length of r = √((-3)² + 5²) = √34 ≈ 5.83, and a direction of θ = arctan(5/-3) ≈ 2.11 radians or 120.96°.

## Finding the Angle in the Correct Quadrant
To find the angle in the correct quadrant, the formula θ = 2 arctan(y/(x + r)) can be used, where x, y, and r are defined as the real component, imaginary component, and length of the complex number, respectively.

## Notations for Polar Form
There are several notations for polar form, including:
- r(cos θ + i sin θ)
- r cis θ
- re^{iθ}
- r∠θ

These notations all represent the same complex number in polar form.

## Multiplying and Dividing Complex Numbers
Multiplying and dividing complex numbers in rectangular form can be complicated, but in polar form, it is much easier. The product of two complex numbers in polar form is the product of their lengths and the sum of their angles: (r₁e^{iθ₁}) × (r₂e^{iθ₂}) = r₁r₂e^{i(θ₁ + θ₂)}. The quotient of two complex numbers in polar form is the quotient of their lengths and the difference of their angles: (r₁e^{iθ₁}) / (r₂e^{iθ₂}) = (r₁/r₂)e^{i(θ₁ - θ₂)}.

## DeMoivre's Theorem
DeMoivre's theorem states that (r(cos x + i sin x))ⁿ = rⁿ(cos nx + i sin nx), where r and x are the length and angle of the complex number, and n is the power to which the complex number is raised. This theorem can be used to find any power or root of a complex number.

## Finding Roots of Complex Numbers
To find the roots of a complex number, the number must first be converted to polar form. Then, the formula (r(cos x + i sin x))^(1/n) = r^(1/n)(cos(x/n + 2πk/n) + i sin(x/n + 2πk/n)) can be used, where k is an integer from 0 to n-1.

## Logarithms of Complex Numbers
The logarithm of a complex number can be found using the formula ln(re^{iθ}) = ln r + iθ. This formula can be used to find the logarithm of any complex number.

## Practice Problems
1. Express in a + bi form:
(a) 62∠240°
(b) 100e^{1.17i}

2. Express in polar form, in both degrees and radian measure:
(a) -42 + 17i
(b) 100i
(c) -14.7

3. Find the three cube roots of -i, in a + bi form.

4. Find (1.04 - 0.10i)^16.

5. Convert the rectangular product and quotient to polar form to verify that they match the polar ones.

6. Find the logarithms, to four decimal places:
(a) -67.12
(b) 1 - i
(c) 734.017e^{0.483i}


# Dictionary: Instantaneous Bandwidth

The instantaneous bandwidth is a seismic attribute calculated using two methods: Barnes (1992) and O'Doherty (1992).

Barnes' equation measures the absolute value of the rate of change of the trace envelope amplitude, where the trace envelope is a key component.

O'Doherty's equation involves the centroid of the power spectrum of a wavelet, variance, and RMS frequency. The centroid frequency is calculated as the frequency corresponding to the centroid of the power spectrum of a wavelet. The variance with respect to the centroid frequency is given by an equation that represents the spread of the power spectrum. The RMS frequency, or second moment of the power spectrum, is calculated using another equation.

Expanding the variance equation shows that it represents the statistical measurements of the power spectrum. These computations are useful attributes that represent the statistics of the seismic wavelet over a time window, making them closely associated with time-smoothed instantaneous attributes. The centroid frequency is the mean frequency with equal energy on either side, while the variance indicates the width of the power spectral density distribution, serving as an indication of spectral bandwidth.

Instantaneous bandwidth is a statistical measure of the seismic wavelet, relating to various physical conditions, including:
- Representing seismic data bandwidth sample by sample as a high-resolution character correlator
- Showing overall effects of absorption and seismic character changes

References:
- Taner, Turhan (1992), Attributes Revisited, Rock Solid Images Houston, Texas (published 2000)


Assuming the correct integrand, with proper grouping, is (t^{2} - 2t)/t^{4} = 1/t^{2} - 2/t^{3}, the integral is -1/t + 1/t^{2} + c.


Each beam in an attack is considered a separate instance of the spell, resulting in a separate damage roll and concentration check. This is different from Magic Missile, which has one damage roll per target.

In a multi-attack spell, each beam that hits triggers a new concentration check.

Assumptions for calculations include:
- 3rd level
- 16 Strength
- Fighting a CR 3 creature
- Average damage is 7.50
- Average crit damage is 12.00
- Attack bonus is +5
- Enemy has AC 13

DPR calculations are based on:
- Crit damage = Crit % x average crit hit damage
- Normal damage = Hit % x average hit damage
- Hit % = 100 – [miss %] – [crit %]

For a Champion Fighter:
- A normal roll has a 35% miss, 10.00% crit, and 55.00% normal hit for a total DPR of 5.33
- A roll with advantage has a 12.25% miss, 19.00% crit, and 68.75% normal hit for a total DPR of 7.44
- A roll with a +2 rule has a 25.00% miss, 10.00% crit, and 65.00% normal hit for a total DPR of 6.08
- A roll with disadvantage has a 42.25% miss, 1.00% crit, and 56.75% normal hit for a total DPR of 4.38
- A roll with a -2 rule has a 45.00% miss, 10.00% crit, and 45.00% normal hit for a total DPR of 4.58

For a Rogue:
- A normal roll has a 35% miss, 5.00% crit, and 60.00% normal hit for a total DPR of 10
- A roll with advantage has a 12.25% miss, 9.75% crit, and 78.00% normal hit for a total DPR of 13.85
- A roll with a +2 rule has a 25.00% miss, 5.00% crit, and 70.00% normal hit for a total DPR of 11.45
- A roll with disadvantage has a 42.25% miss, 1.00% crit, and 56.75% normal hit for a total DPR of 4.38
- A roll with a -2 rule has a 45.00% miss, 5.00% crit, and 50.00% normal hit for a total DPR of 3.45

The proposed houserule decreases the expected DPR for any class, especially those that rely on burst damage. The chance of a critical hit does not change, but the overall expected damage output of the party is reduced.

AnyDice can be used to calculate statistics, storing dice sequences in variables and considering every possible outcome. The program can calculate the chance to hit, damage, and other statistics for different attacks and rules.

The chance to hit with shocking grasp is 55%, and the chance to hit with at least one of four Eldritch Blast attacks with disadvantage is 38.78% higher. The average damage for shocking grasp is 4.95, and for Eldritch Blast, it is calculated separately for each attack.

With advantage, the chance to hit with shocking grasp increases to 79.75%, making it almost equal to the chance of hitting with Eldritch Blast. The average damage increases by around 33% compared to all-disadvantage.

The program can be used to calculate various statistics and outcomes for different attacks and rules, providing a useful tool for analyzing and comparing different scenarios.

What is the expected DPR for a Champion Fighter with a +2 rule?
A) 5.33
B) 6.08
C) 7.44
D) 4.38

What is the expected DPR for a Rogue with advantage?
A) 10
B) 11.45
C) 13.85
D) 4.38

What is the increase in average damage for Eldritch Blast with advantage compared to all-disadvantage?
A) 25%
B) 33%
C) 50%
D) 75%


A walking droplet in a small 1D cavity exhibits "random" motion when the system's memory is high enough, i.e., when the forcing is strong enough but still below the Faraday Threshold. The droplet is placed in a square box at the onset of the Faraday threshold, and its trajectory is mapped.

In the long-time limit, a self-interference pattern appears, and its shape relates to the square cavity surface wave eigen-modes. This experiment aims to reproduce the work of Bush et al. in a square box. The results show that a walking droplet in a square cavity displays random motion, but over time, its trajectory builds a statistic reminiscent of the resonant mode of the cavity.

The position distribution, or probability density, of the droplet is computed, and the temporal resolution of the measurement is 240 fps at 848×480 pixels. The forcing frequency is 60 Hz, and the Faraday frequency is 30 Hz, allowing for observation of the dynamic without any strobe effect.

A simulation of a dotwave in a harmonic potential, with no viscous dissipation, exhibits successive clockwise and anti-clockwise circle-like motions, separated by intermittences with null mean angular momentum. The simulation integrates the motion equation continuously via a MATLAB delay differential equation solver.

The dot "reads" the value of the field continuously and "writes" to the field every T_F, creating a local wave field represented by a Bessel function, which is then slowly damped. Interferences between the waves created by the last 300 bounces are computed at each integration step to obtain the shape of the wave and the motion of the dot.

In a small cavity, approximately 6 to 8 times the Faraday wavelength, the walking droplet exhibits several normal modes of vibration. Initially, the movement seems "random," but the whole wave-particle system synchronizes and starts turning altogether. Different modes are possible, with different radii for the droplet trajectory, and these radii coincide with the knots of the Faraday standing waves of the cavity, resulting in a quantization of the radius of the different trajectories.

Question: What is the (m, n) mode of the first droplet shown in the movie?