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-<s>[INST]How many sides does a triangle have?[/INST]A triangle has three sides.</s>
-<s>[INST]What is the sum of the interior angles of a triangle?[/INST]The sum of the interior angles of a triangle is 180 degrees.</s>
-<s>[INST]Can a triangle have more than one right angle?[/INST]No, a triangle cannot have more than one right angle.</s>
-<s>[INST]What is an equilateral triangle?[/INST]An equilateral triangle has three equal sides and three equal angles.</s>
-<s>[INST]What is an isosceles triangle?[/INST]An isosceles triangle has at least two equal sides and two equal angles.</s>
-<s>[INST]What is a scalene triangle?[/INST]A scalene triangle has all sides of different lengths and all angles of different measures.</s>
-<s>[INST]What is an acute triangle?[/INST]An acute triangle has all three angles less than 90 degrees.</s>
-<s>[INST]What is a right triangle?[/INST]A right triangle has one angle equal to 90 degrees.</s>
-<s>[INST]What is an obtuse triangle?[/INST]An obtuse triangle has one angle greater than 90 degrees.</s>
-<s>[INST]How do you calculate the area of a triangle?[/INST]The area of a triangle is calculated as 0.5 * base * height.</s>
-<s>[INST]How do you calculate the perimeter of a triangle?[/INST]The perimeter of a triangle is the sum of its three sides.</s>
-<s>[INST]What is the hypotenuse of a right triangle?[/INST]The hypotenuse is the longest side of a right triangle, opposite the right angle.</s>
-<s>[INST]What is the Pythagorean theorem?[/INST]The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.</s>
-<s>[INST]What are the different types of triangles based on angles?[/INST]The types of triangles based on angles are acute, right, and obtuse triangles.</s>
-<s>[INST]What are the different types of triangles based on sides?[/INST]The types of triangles based on sides are equilateral, isosceles, and scalene triangles.</s>
-<s>[INST]What is an exterior angle of a triangle?[/INST]An exterior angle of a triangle is formed by one side of the triangle and the extension of an adjacent side.</s>
-<s>[INST]What is the relationship between an exterior angle and its interior opposite angles?[/INST]An exterior angle of a triangle is equal to the sum of the opposite interior angles.</s>
-<s>[INST]Can a triangle be both isosceles and obtuse?[/INST]Yes, a triangle can be both isosceles and obtuse.</s>
-<s>[INST]Can a triangle be both scalene and right?[/INST]Yes, a triangle can be both scalene and right.</s>
-<s>[INST]What is the centroid of a triangle?[/INST]The centroid of a triangle is the point where its three medians intersect.</s>
-<s>[INST]What is the circumcenter of a triangle?[/INST]The circumcenter is the point where the perpendicular bisectors of the sides of a triangle intersect.</s>
-<s>[INST]What is the incenter of a triangle?[/INST]The incenter is the point where the angle bisectors of a triangle intersect.</s>
-<s>[INST]What is the orthocenter of a triangle?[/INST]The orthocenter is the point where the altitudes of a triangle intersect.</s>
-<s>[INST]What is a median of a triangle?[/INST]A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side.</s>
-<s>[INST]What is an altitude of a triangle?[/INST]An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side.</s>
-<s>[INST]What is an angle bisector of a triangle?[/INST]An angle bisector of a triangle is a line segment that bisects one of the vertex angles of the triangle.</s>
-<s>[INST]How do you find the length of a side in a right triangle?[/INST]In a right triangle, the length of a side can be found using the Pythagorean theorem.</s>
-<s>[INST]What is the formula for the area of an equilateral triangle?[/INST]The area of an equilateral triangle is (sqrt(3)/4) * side^2.</s>
-<s>[INST]What is Heron's formula?[/INST]Heron's formula is used to find the area of a triangle when the lengths of all three sides are known.</s>
-<s>[INST]What is the sine rule for triangles?[/INST]The sine rule states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant.</s>
-<s>[INST]What is the cosine rule for triangles?[/INST]The cosine rule states that in any triangle, the square of a side is equal to the sum of the squares of the other two sides minus twice the product of the other two sides and the cosine of the included angle.</s>
-<s>[INST]How can you tell if three lengths can form a triangle?[/INST]Three lengths can form a triangle if the sum of any two lengths is greater than the third length.</s>
-<s>[INST]What is the triangle inequality theorem?[/INST]The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side.</s>
-<s>[INST]What are similar triangles?[/INST]Similar triangles have the same shape but not necessarily the same size, with corresponding angles equal and corresponding sides proportional.</s>
-<s>[INST]What are congruent triangles?[/INST]Congruent triangles have the same shape and size, with corresponding sides and angles equal.</s>
-<s>[INST]How can you prove that two triangles are congruent?[/INST]Two triangles are congruent if they satisfy the criteria: SSS, SAS, ASA, AAS, or HL.</s>
-<s>[INST]How can you prove that two triangles are similar?[/INST]Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional.</s>
-<s>[INST]What is a right-angled triangle?[/INST]A right-angled triangle has one angle equal to 90 degrees.</s>
-<s>[INST]What is the smallest angle in an equilateral triangle?[/INST]In an equilateral triangle, all angles are equal to 60 degrees.</s>
-<s>[INST]Can an isosceles triangle be equilateral?[/INST]Yes, if an isosceles triangle has all three sides equal, it is also equilateral.</s>
-<s>[INST]What is a right-angled isosceles triangle?[/INST]A right-angled isosceles triangle has a right angle and two equal sides.</s>
-<s>[INST]What is the largest angle in an equilateral triangle?[/INST]In an equilateral triangle, all angles are 60 degrees.</s>
-<s>[INST]How do you draw a triangle given three sides?[/INST]To draw a triangle given three sides, use a ruler and compass to construct it step by step.</s>
-<s>[INST]How do you draw a triangle given one side and two angles?[/INST]To draw a triangle given one side and two angles, use a protractor to measure the angles and a ruler for the side.</s>
-<s>[INST]What is the circumcircle of a triangle?[/INST]The circumcircle of a triangle is the circle that passes through all three vertices of the triangle.</s>
-<s>[INST]What is the incircle of a triangle?[/INST]The incircle of a triangle is the circle that is tangent to all three sides of the triangle.</s>
-<s>[INST]What is a degenerate triangle?[/INST]A degenerate triangle has all three vertices on a single line.</s>
-<s>[INST]What are the properties of a triangle?[/INST]The properties of a triangle include having three sides, three angles, the sum of interior angles being 180 degrees, and the sum of exterior angles being 360 degrees.</s>
-<s>[INST]What is a triangle's vertex?[/INST]A vertex of a triangle is a point where two sides of the triangle meet.</s>
-<s>[INST]How do you determine if a triangle is obtuse?[/INST]A triangle is obtuse if one of its interior angles is greater than 90 degrees.</s>
-<s>[INST]What is the Euler line in a triangle?[/INST]The Euler line is a line that goes through several important points in a triangle, including the centroid, orthocenter, and circumcenter.</s>
-<s>[INST]What is the nine-point circle of a triangle?[/INST]The nine-point circle passes through nine significant concurrent points in a triangle, including the midpoints of the sides and the feet of the altitudes.</s>
-<s>[INST]What are the basic properties of a triangle?[/INST]The basic properties of a triangle include having three sides, three angles, and the sum of the interior angles is always 180 degrees.</s>
-<s>[INST]What is the area of a right triangle with base 3 and height 4?[/INST]The area of a right triangle with base 3 and height 4 is 6 square units.</s>
-<s>[INST]Can a triangle be both acute and equilateral?[/INST]Yes, if all angles are less than 90 degrees and all sides are equal, the triangle is both acute and equilateral.</s>
-<s>[INST]What is a right-angled scalene triangle?[/INST]A right-angled scalene triangle has one right angle and no equal sides.</s>
-<s>[INST]What is the formula for the height of an equilateral triangle?[/INST]The height of an equilateral triangle is given by the formula (sqrt(3)/2) * side.</s>
-<s>[INST]What are the interior angles of an equilateral triangle?[/INST]The interior angles of an equilateral triangle are each 60 degrees.</s>
-<s>[INST]Can an obtuse triangle be isosceles?[/INST]Yes, an obtuse triangle can also be isosceles if it has two equal sides and one angle greater than 90 degrees.</s>
-<s>[INST]What is the longest side of a right triangle called?[/INST]The longest side of a right triangle is called the hypotenuse.</s>
-<s>[INST]How do you find the length of the hypotenuse in a right triangle?[/INST]The length of the hypotenuse in a right triangle is found using the Pythagorean theorem: hypotenuse^2 = base^2 + height^2.</s>
-<s>[INST]What is a 45-45-90 triangle?[/INST]A 45-45-90 triangle is a right triangle with two 45 degree angles and one 90 degree angle.</s>
-<s>[INST]What is a 30-60-90 triangle?[/INST]A 30-60-90 triangle is a right triangle with one 30 degree angle, one 60 degree angle, and one 90 degree angle.</s>
-<s>[INST]How do you construct a triangle given two sides and the included angle?[/INST]To construct a triangle given two sides and the included angle, use a compass and straightedge to replicate the sides and angle.</s>
-<s>[INST]What is the orthic triangle?[/INST]The orthic triangle is formed by the feet of the altitudes of a given triangle.</s>
-<s>[INST]What is the medial triangle?[/INST]The medial triangle is formed by connecting the midpoints of the sides of the original triangle.</s>
-<s>[INST]What is a degenerate triangle?[/INST]A degenerate triangle has all three vertices on a single line, essentially forming a straight line.</s>
-<s>[INST]What is the exterior angle theorem?[/INST]The exterior angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.</s>
-<s>[INST]What is the altitude of a right triangle?[/INST]The altitude of a right triangle is a perpendicular segment from the right angle vertex to the hypotenuse.</s>
-<s>[INST]What is the area formula for a triangle using trigonometry?[/INST]The area formula for a triangle using trigonometry is 0.5 * a * b * sin(C), where a and b are sides and C is the included angle.</s>
-<s>[INST]How do you find the circumradius of a triangle?[/INST]The circumradius of a triangle can be found using the formula R = (abc) / (4K), where a, b, and c are the side lengths and K is the area.</s>
-<s>[INST]What is the incentre of a triangle?[/INST]The incenter is the point where the angle bisectors of a triangle intersect, and it is the center of the triangle's incircle.</s>
-<s>[INST]What is a cevian in a triangle?[/INST]A cevian is a line segment from a vertex of a triangle to the opposite side (or its extension).</s>
-<s>[INST]What is the relationship between the sides of a 30-60-90 triangle?[/INST]In a 30-60-90 triangle, the sides are in the ratio 1:sqrt(3):2.</s>
-<s>[INST]What is the relationship between the sides of a 45-45-90 triangle?[/INST]In a 45-45-90 triangle, the sides are in the ratio 1:1:sqrt(2).</s>
-<s>[INST]How do you prove that the sum of the interior angles of a triangle is 180 degrees?[/INST]To prove that the sum of the interior angles of a triangle is 180 degrees, draw a parallel line through one vertex and use alternate interior angles.</s>
-<s>[INST]What is the Fermat point of a triangle?[/INST]The Fermat point of a triangle is the point such that the total distance to the three vertices is minimized.</s>
-<s>[INST]What is the Steiner point in a triangle?[/INST]The Steiner point is another point that minimizes the total distance to the three vertices, used in network design.</s>
-<s>[INST]What is a Viviani's theorem?[/INST]Viviani's theorem states that the sum of the perpendicular distances from any interior point to the sides of an equilateral triangle is constant.</s>
-<s>[INST]How do you determine if three points form a triangle?[/INST]Three points form a triangle if they are not collinear.</s>
-<s>[INST]What is the angle bisector theorem?[/INST]The angle bisector theorem states that the angle bisector of a triangle divides the opposite side into segments proportional to the adjacent sides.</s>
-<s>[INST]What is the excenter of a triangle?[/INST]The excenter of a triangle is the center of an excircle, which is a circle tangent to one side and the extensions of the other two sides.</s>
-<s>[INST]How do you calculate the exradius of a triangle?[/INST]The exradius is calculated using the formula r = A/s, where A is the area of the triangle and s is the semiperimeter.</s>
-<s>[INST]What is a Napoleon triangle?[/INST]A Napoleon triangle is formed by constructing equilateral triangles on the sides of any given triangle.</s>
-<s>[INST]What is a Morley triangle?[/INST]A Morley triangle is formed by the intersections of the adjacent trisectors of the angles of a triangle.</s>
-<s>[INST]What is a pedal triangle?[/INST]A pedal triangle is formed by projecting a point onto the sides of a reference triangle.</s>
-<s>[INST]What is a contact triangle?[/INST]A contact triangle is formed by the points where the incircle or excircles touch the sides of the triangle.</s>
-<s>[INST]What is a Schmidt triangle?[/INST]A Schmidt triangle is formed by drawing perpendiculars from the vertices of a triangle to the opposite sides.</s>
-<s>[INST]What is a Nagel point?[/INST]The Nagel point is the point where the triangle's bisectors of the sides and the angle bisectors of the angles meet.</s>
-<s>[INST]What is the Routh's theorem?[/INST]Routh's theorem provides a relation between the areas of a triangle and its cevian triangle.</s>
-<s>[INST]What is the van Aubel's theorem?[/INST]Van Aubel's theorem states that if squares are constructed on the sides of a quadrilateral, the line segments joining the centers of opposite squares are equal and perpendicular.</s>
-<s>[INST]What is the Gergonne point?[/INST]The Gergonne point is the point where the cevians that connect the vertices to the points where the incircle touches the opposite sides intersect.</s>
-<s>[INST]What is the Spieker center?[/INST]The Spieker center is the center of the circle that is tangent to the midpoints of the sides of the triangle.</s>
-<s>[INST]What is the Miquel point?[/INST]The Miquel point is the point of concurrency of the circles passing through the vertices of a triangle and the feet of the cevians.</s>
-<s>[INST]What is the Feuerbach point?[/INST]The Feuerbach point is the point where the incircle and the nine-point circle of a triangle touch.</s>
-<s>[INST]What is a Poristic triangle?[/INST]A Poristic triangle is a triangle where the incircle and circumcircle are concentric.</s>
-<s>[INST]What is the Euler's triangle inequality?[/INST]Euler's triangle inequality states that the distance between the circumcenter and the incenter is less than or equal to the radius of the circumcircle minus the radius of the incircle.</s>
-<s>[INST]What is the Cevian triangle?[/INST]The Cevian triangle is formed by the cevians from each vertex of the original triangle.</s>
-<s>[INST]What is a triangle?[/INST]A triangle is a polygon with three edges and three vertices.</s>
-<s>[INST]How many sides does a triangle have?[/INST]A triangle has three sides.</s>
-<s>[INST]What is the sum of the interior angles of a triangle?[/INST]The sum of the interior angles of a triangle is 180 degrees.</s>
-<s>[INST]Can a triangle have more than one right angle?[/INST]No, a triangle cannot have more than one right angle.</s>
-<s>[INST]What is an equilateral triangle?[/INST]An equilateral triangle has three equal sides and three equal angles.</s>
-<s>[INST]What is an isosceles triangle?[/INST]An isosceles triangle has at least two equal sides and two equal angles.</s>
-<s>[INST]What is a scalene triangle?[/INST]A scalene triangle has all sides of different lengths and all angles of different measures.</s>
-<s>[INST]What is an acute triangle?[/INST]An acute triangle has all three angles less than 90 degrees.</s>
-<s>[INST]What is a right triangle?[/INST]A right triangle has one angle equal to 90 degrees.</s>
-<s>[INST]What is an obtuse triangle?[/INST]An obtuse triangle has one angle greater than 90 degrees.</s>
-<s>[INST]How do you calculate the area of a triangle?[/INST]The area of a triangle is calculated as 0.5 * base * height.</s>
-<s>[INST]How do you calculate the perimeter of a triangle?[/INST]The perimeter of a triangle is the sum of its three sides.</s>
-<s>[INST]What is the hypotenuse of a right triangle?[/INST]The hypotenuse is the longest side of a right triangle, opposite the right angle.</s>
-<s>[INST]What is the Pythagorean theorem?[/INST]The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.</s>
-<s>[INST]What are the different types of triangles based on angles?[/INST]The types of triangles based on angles are acute, right, and obtuse triangles.</s>
-<s>[INST]What are the different types of triangles based on sides?[/INST]The types of triangles based on sides are equilateral, isosceles, and scalene triangles.</s>
-<s>[INST]What is an exterior angle of a triangle?[/INST]An exterior angle of a triangle is formed by one side of the triangle and the extension of an adjacent side.</s>
-<s>[INST]What is the relationship between an exterior angle and its interior opposite angles?[/INST]An exterior angle of a triangle is equal to the sum of the opposite interior angles.</s>
-<s>[INST]Can a triangle be both isosceles and obtuse?[/INST]Yes, a triangle can be both isosceles and obtuse.</s>
-<s>[INST]Can a triangle be both scalene and right?[/INST]Yes, a triangle can be both scalene and right.</s>
-<s>[INST]What is the centroid of a triangle?[/INST]The centroid of a triangle is the point where its three medians intersect.</s>
-<s>[INST]What is the circumcenter of a triangle?[/INST]The circumcenter is the point where the perpendicular bisectors of the sides of a triangle intersect.</s>
-<s>[INST]What is the incenter of a triangle?[/INST]The incenter is the point where the angle bisectors of a triangle intersect.</s>
-<s>[INST]What is the orthocenter of a triangle?[/INST]The orthocenter is the point where the altitudes of a triangle intersect.</s>
-<s>[INST]What is a median of a triangle?[/INST]A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side.</s>
-<s>[INST]What is an altitude of a triangle?[/INST]An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side.</s>
-<s>[INST]What is an angle bisector of a triangle?[/INST]An angle bisector of a triangle is a line segment that bisects one of the vertex angles of the triangle.</s>
-<s>[INST]How do you find the length of a side in a right triangle?[/INST]In a right triangle, the length of a side can be found using the Pythagorean theorem.</s>
-<s>[INST]What is the formula for the area of an equilateral triangle?[/INST]The area of an equilateral triangle is (sqrt(3)/4) * side^2.</s>
-<s>[INST]What is Heron's formula?[/INST]Heron's formula is used to find the area of a triangle when the lengths of all three sides are known.</s>
-<s>[INST]What is the sine rule for triangles?[/INST]The sine rule states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant.</s>
-<s>[INST]What is the cosine rule for triangles?[/INST]The cosine rule states that in any triangle, the square of a side is equal to the sum of the squares of the other two sides minus twice the product of the other two sides and the cosine of the included angle.</s>
-<s>[INST]How can you tell if three lengths can form a triangle?[/INST]Three lengths can form a triangle if the sum of any two lengths is greater than the third length.</s>
-<s>[INST]What is the triangle inequality theorem?[/INST]The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side.</s>
-<s>[INST]What are similar triangles?[/INST]Similar triangles have the same shape but not necessarily the same size, with corresponding angles equal and corresponding sides proportional.</s>
-<s>[INST]What are congruent triangles?[/INST]Congruent triangles have the same shape and size, with corresponding sides and angles equal.</s>
-<s>[INST]How can you prove that two triangles are congruent?[/INST]Two triangles are congruent if they satisfy the criteria: SSS, SAS, ASA, AAS, or HL.</s>
-<s>[INST]How can you prove that two triangles are similar?[/INST]Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional.</s>
-<s>[INST]What is a right-angled triangle?[/INST]A right-angled triangle has one angle equal to 90 degrees.</s>
-<s>[INST]What is the smallest angle in an equilateral triangle?[/INST]In an equilateral triangle, all angles are equal to 60 degrees.</s>
-<s>[INST]Can an isosceles triangle be equilateral?[/INST]Yes, if an isosceles triangle has all three sides equal, it is also equilateral.</s>
-<s>[INST]What is a right-angled isosceles triangle?[/INST]A right-angled isosceles triangle has a right angle and two equal sides.</s>
-<s>[INST]What is the largest angle in an equilateral triangle?[/INST]In an equilateral triangle, all angles are 60 degrees.</s>
-<s>[INST]How do you draw a triangle given three sides?[/INST]To draw a triangle given three sides, use a ruler and compass to construct it step by step.</s>
-<s>[INST]How do you draw a triangle given one side and two angles?[/INST]To draw a triangle given one side and two angles, use a protractor to measure the angles and a ruler for the side.</s>
-<s>[INST]What is the circumcircle of a triangle?[/INST]The circumcircle of a triangle is the circle that passes through all three vertices of the triangle.</s>
-<s>[INST]What is the incircle of a triangle?[/INST]The incircle of a triangle is the circle that is tangent to all three sides of the triangle.</s>
-<s>[INST]What is a degenerate triangle?[/INST]A degenerate triangle has all three vertices on a single line.</s>
-<s>[INST]What are the properties of a triangle?[/INST]The properties of a triangle include having three sides, three angles, the sum of interior angles being 180 degrees, and the sum of exterior angles being 360 degrees.</s>
-<s>[INST]What is a triangle's vertex?[/INST]A vertex of a triangle is a point where two sides of the triangle meet.</s>
-<s>[INST]How do you determine if a triangle is obtuse?[/INST]A triangle is obtuse if one of its interior angles is greater than 90 degrees.</s>
-<s>[INST]What is the Euler line in a triangle?[/INST]The Euler line is a line that goes through several important points in a triangle, including the centroid, orthocenter, and circumcenter.</s>
-<s>[INST]What is the nine-point circle of a triangle?[/INST]The nine-point circle passes through nine significant concurrent points in a triangle, including the midpoints of the sides and the feet of the altitudes.</s>
-<s>[INST]What are the basic properties of a triangle?[/INST]The basic properties of a triangle include having three sides, three angles, and the sum of the interior angles is always 180 degrees.</s>
-<s>[INST]What is the area of a right triangle with base 3 and height 4?[/INST]The area of a right triangle with base 3 and height 4 is 6 square units.</s>
-<s>[INST]Can a triangle be both acute and equilateral?[/INST]Yes, if all angles are less than 90 degrees and all sides are equal, the triangle is both acute and equilateral.</s>
-<s>[INST]What is a right-angled scalene triangle?[/INST]A right-angled scalene triangle has one right angle and no equal sides.</s>
-<s>[INST]What is the formula for the height of an equilateral triangle?[/INST]The height of an equilateral triangle is given by the formula (sqrt(3)/2) * side.</s>
-<s>[INST]What are the interior angles of an equilateral triangle?[/INST]The interior angles of an equilateral triangle are each 60 degrees.</s>
-<s>[INST]Can an obtuse triangle be isosceles?[/INST]Yes, an obtuse triangle can also be isosceles if it has two equal sides and one angle greater than 90 degrees.</s>
-<s>[INST]What is the longest side of a right triangle called?[/INST]The longest side of a right triangle is called the hypotenuse.</s>
-<s>[INST]How do you find the length of the hypotenuse in a right triangle?[/INST]The length of the hypotenuse in a right triangle is found using the Pythagorean theorem: hypotenuse^2 = base^2 + height^2.</s>
-<s>[INST]What is a 45-45-90 triangle?[/INST]A 45-45-90 triangle is a right triangle with two 45 degree angles and one 90 degree angle.</s>
-<s>[INST]What is a 30-60-90 triangle?[/INST]A 30-60-90 triangle is a right triangle with one 30 degree angle, one 60 degree angle, and one 90 degree angle.</s>
-<s>[INST]How do you construct a triangle given two sides and the included angle?[/INST]To construct a triangle given two sides and the included angle, use a compass and straightedge to replicate the sides and angle.</s>
-<s>[INST]What is the orthic triangle?[/INST]The orthic triangle is formed by the feet of the altitudes of a given triangle.</s>
-<s>[INST]What is the medial triangle?[/INST]The medial triangle is formed by connecting the midpoints of the sides of the original triangle.</s>
-<s>[INST]What is a degenerate triangle?[/INST]A degenerate triangle has all three vertices on a single line, essentially forming a straight line.</s>
-<s>[INST]What is the exterior angle theorem?[/INST]The exterior angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.</s>
-<s>[INST]What is the altitude of a right triangle?[/INST]The altitude of a right triangle is a perpendicular segment from the right angle vertex to the hypotenuse.</s>
-<s>[INST]What is the area formula for a triangle using trigonometry?[/INST]The area formula for a triangle using trigonometry is 0.5 * a * b * sin(C), where a and b are sides and C is the included angle.</s>
-<s>[INST]How do you find the circumradius of a triangle?[/INST]The circumradius of a triangle can be found using the formula R = (abc) / (4K), where a, b, and c are the side lengths and K is the area.</s>
-<s>[INST]What is the incentre of a triangle?[/INST]The incenter is the point where the angle bisectors of a triangle intersect, and it is the center of the triangle's incircle.</s>
-<s>[INST]What is a cevian in a triangle?[/INST]A cevian is a line segment from a vertex of a triangle to the opposite side (or its extension).</s>
-<s>[INST]What is the relationship between the sides of a 30-60-90 triangle?[/INST]In a 30-60-90 triangle, the sides are in the ratio 1:sqrt(3):2.</s>
-<s>[INST]What is the relationship between the sides of a 45-45-90 triangle?[/INST]In a 45-45-90 triangle, the sides are in the ratio 1:1:sqrt(2).</s>
-<s>[INST]How do you prove that the sum of the interior angles of a triangle is 180 degrees?[/INST]To prove that the sum of the interior angles of a triangle is 180 degrees, draw a parallel line through one vertex and use alternate interior angles.</s>
-<s>[INST]What is the Fermat point of a triangle?[/INST]The Fermat point of a triangle is the point such that the total distance to the three vertices is minimized.</s>
-<s>[INST]What is the Steiner point in a triangle?[/INST]The Steiner point is another point that minimizes the total distance to the three vertices, used in network design.</s>
-<s>[INST]What is a Viviani's theorem?[/INST]Viviani's theorem states that the sum of the perpendicular distances from any interior point to the sides of an equilateral triangle is constant.</s>
-<s>[INST]How do you determine if three points form a triangle?[/INST]Three points form a triangle if they are not collinear.</s>
-<s>[INST]What is the angle bisector theorem?[/INST]The angle bisector theorem states that the angle bisector of a triangle divides the opposite side into segments proportional to the adjacent sides.</s>
-<s>[INST]What is the excenter of a triangle?[/INST]The excenter of a triangle is the center of an excircle, which is a circle tangent to one side and the extensions of the other two sides.</s>
-<s>[INST]How do you calculate the exradius of a triangle?[/INST]The exradius is calculated using the formula r = A/s, where A is the area of the triangle and s is the semiperimeter.</s>
-<s>[INST]What is a Napoleon triangle?[/INST]A Napoleon triangle is formed by constructing equilateral triangles on the sides of any given triangle.</s>
-<s>[INST]What is a Morley triangle?[/INST]A Morley triangle is formed by the intersections of the adjacent trisectors of the angles of a triangle.</s>
-<s>[INST]What is a pedal triangle?[/INST]A pedal triangle is formed by projecting a point onto the sides of a reference triangle.</s>
-<s>[INST]What is a contact triangle?[/INST]A contact triangle is formed by the points where the incircle or excircles touch the sides of the triangle.</s>
-<s>[INST]What is a Schmidt triangle?[/INST]A Schmidt triangle is formed by drawing perpendiculars from the vertices of a triangle to the opposite sides.</s>
-<s>[INST]What is a Nagel point?[/INST]The Nagel point is the point where the triangle's bisectors of the sides and the angle bisectors of the angles meet.</s>
-<s>[INST]What is the Routh's theorem?[/INST]Routh's theorem provides a relation between the areas of a triangle and its cevian triangle.</s>
-<s>[INST]What is the van Aubel's theorem?[/INST]Van Aubel's theorem states that if squares are constructed on the sides of a quadrilateral, the line segments joining the centers of opposite squares are equal and perpendicular.</s>
-<s>[INST]What is the Gergonne point?[/INST]The Gergonne point is the point where the cevians that connect the vertices to the points where the incircle touches the opposite sides intersect.</s>
-<s>[INST]What is the Spieker center?[/INST]The Spieker center is the center of the circle that is tangent to the midpoints of the sides of the triangle.</s>
-<s>[INST]What is the Miquel point?[/INST]The Miquel point is the point of concurrency of the circles passing through the vertices of a triangle and the feet of the cevians.</s>
-<s>[INST]What is the Feuerbach point?[/INST]The Feuerbach point is the point where the incircle and the nine-point circle of a triangle touch.</s>
-<s>[INST]What is a Poristic triangle?[/INST]A Poristic triangle is a triangle where the incircle and circumcircle are concentric.</s>
-<s>[INST]What is the Euler's triangle inequality?[/INST]Euler's triangle inequality states that the distance between the circumcenter and the incenter is less than or equal to the radius of the circumcircle minus the radius of the incircle.</s>
-<s>[INST]What is the Cevian triangle?[/INST]The Cevian triangle is formed by the cevians from each vertex of the original triangle.</s>
-<s>[INST]who give permission to ram?[/INST]'Bikram' the man behind the story give permission to ram.</s>
-<s>[INST]What teach math to ayush[/INST]'Bikram' the mathematician teach math to ayush.</s>