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4.3. Two balls of one radius and two of another are arranged so that each ball touches three others and a given plane. Find the ratio of the radii of the balls. | 4.3. Let $A$ and $C$ be the points of tangency of spheres of radius $R$ with the plane; $B$ and $D$ be the points of tangency of spheres of radius $r$ with the plane. According to problem 4.1, $A B = B C = C D = A D = 2 \sqrt{R r}$, so $A B C D$ is a rhombus; its diagonals are equal to $2 R$ and $2 r$. Therefore, $R^{2... | 2+\sqrt{3} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,147 |
4.4. The radii of two non-intersecting spheres are $R$ and $r$; the distance between their centers is $a$. Within what limits can the length of the common tangent to these spheres vary? | 4.4. Let $M N$ be the common tangent, $A$ and $B$ be the centers of the spheres. The radii $A M$ and $B N$ are perpendicular to the tangent $M N$. Let $C$ be the projection of point $A$ onto the plane passing through point $N$ perpendicular to $M N$ (Fig. 35). Since $N B=r$ and $N C=R$, $B C$ can vary from $R+r$ to $|R... | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,148 | |
4.5. Two tangent spheres are inscribed in a dihedral angle of magnitude $2 \alpha$. Let $A$ be the point of tangency of the first sphere with the first face, and $B$ be the point of tangency of the second sphere with the second face. In what ratio is the segment $A B$ divided by the points of intersection with these sp... | 4.5. Let $a$ and $b$ be the radii of the spheres, $A_{1}$ and $B_{\mathrm{i}}$ be the points of their tangency with the faces of the angle. The sides of the trapezoid $A A, B B_{1}$ can be easily calculated: $A B_{1}=A_{1} B=2 \sqrt{a b}$ (Problem 4.1), $A A_{1}=$
$=2 a \cos \alpha$ and $B B_{1}=2 b \cos \alpha$. The s... | \cos^{2}\alpha:\sin^{2}\alpha:\cos^{2}\alpha | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,149 |
4.6. Perpendiculars are dropped from an arbitrary point in space to the planes of the faces of a given cube. The resulting segments are the diagonals of six other cubes. Consider six spheres, each of which touches all the edges of the corresponding cube. Prove that all these spheres have a common tangent line. | 4.6. Let's first consider the cube $A B C D A_{1} B_{1} C_{1} D_{1}$. The cone with axis $A C_{1}$ and generatrix $A B$ touches the sphere that touches all the edges of the given cube. Therefore, the cone with axis $A B$ and generatrix $A C_{1}$ touches the sphere that touches all the edges of the cube with diagonal $A... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,150 |
4.7. A sphere with diameter $C E$ touches the plane $A B C$ at point $C$; $A D$ is a tangent to this sphere. Prove that if point $B$ lies on the line $D E$, then $A C=A B$. | 4.7. Since $A C$ and $A D$ are tangents to the given sphere, they are equal. Therefore, point $A$ lies in the plane passing through the midpoint of segment $C D$ and perpendicular to it. Since $\angle C D B=90^{\circ}$, this plane intersects the plane $A B C$ along a line passing through the midpoint of segment $B C$ a... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,151 |
4.8. Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$. A plane passing through vertex $A$ and tangent to the sphere inscribed in the cube intersects the edges $A_{1} B_{1}$ and $A_{1} D_{1}$ at points $K$ and $N$,
Find the measure of the angle between the planes $A C_{1} K$ and $A C_{1} N$. | 4.8. Let us first prove the following auxiliary statement. Let two planes, intersecting along the line $A X$, touch a sphere with center $O$ at points $F$ and $G$. Then $A O X$ is the bisector plane of the dihedral angle formed by the planes $A O F$ and $A O G$. Indeed, the points $F$ and $G$ are symmetric relative to ... | 60 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,152 |
4.9. Two equal triangles $K L M$ and $K L N$ share a common side $K L$, and $\angle K L M=\angle L K N=60^{\circ}$, $K L=1$ and $L M=K N=6$. The planes $K L M$ and $K L N$ are perpendicular. Find the radius of the sphere that touches the segments $L M$ and $K N$ at their midpoints. | 4.9. Let $O_{1}$ and $O_{2}$ be the projections of the center $O$ of a given sphere onto the planes $K L M$ and $K L N$; $P$ and $S$ are the midpoints of the segments $L M$ and $K N$. Since $O P=O S$ and $P K=S L$, then $O K=O L$. Therefore, the projection of points $O_{1}$ and $O_{2}$ onto the line $K L$ is the point ... | \frac{137}{12} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,153 |
4.10. Tangents are drawn from points $A$ and $B$ to a given sphere. Prove that all points of their intersection, distinct from $A$ and $B$, lie in two planes. | 4.10. Let $O$ - the center of a given sphere, $r$ - its radius; $a$ and $b$ - the lengths of the tangents drawn from points $A$ and $B$; $M$ the point of intersection of the tangents drawn from $A$ and $B$; $x$ the length of the tangent drawn from $M$. Then $A M^{2}=(a \pm x)^{2}$, $B M^{2}=(b \pm x)^{2}$ and $O M^{2}=... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,154 |
4.11. The centers of three spheres, with radii of 3, 4, and 6, are located at the vertices of an equilateral triangle with a side length of 11. How many planes exist that are tangent to all three spheres simultaneously?
## § 3. Two intersecting circles lie on the same sphere | 4.11. Consider a plane tangent to all three given spheres, and draw a plane through the center of the sphere of radius 3 parallel to it. The resulting plane is tangent to the spheres of radii $4 \pm 3$ and $6 \pm 3$, concentric with the spheres of radii 4 and 6. When the signs of the number 3 are the same, the tangency... | 3 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,155 |
4.12. a) Two circles, not lying in the same plane, intersect at two distinct points $A$ and $B$. Prove that there exists a unique sphere containing these circles.
b) Two circles, not lying in the same plane, touch a line $l$ at point $P$. Prove that there exists a unique sphere containing these circles. | 4.12. Let $O_{\mathbf{i}}$ and $O_{2}$ be the centers of the given circles: in problem a) $M$ is the midpoint of segment $A B$, in problem b) $M=P$. Consider the plane $M O_{1} O_{2}$. The center of the desired sphere is the point of intersection of the perpendiculars erected in this plane from points $O_{1}$ and $O_{2... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,156 |
4.13. Given a truncated triangular pyramid. Prove that if two of its lateral faces are inscribed quadrilaterals, then the third lateral face is also an inscribed quadrilateral. | 4.13. The circumscribed circles of two lateral faces have two common points - the common vertices of these faces. Therefore, there exists a sphere that contains both of these circles. The circumscribed circle of the third face is the section of that sphere by the plane of the face. | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,157 |
4.14. All faces of a convex polyhedron are convex polygons, and all angles are trihedral. Prove that a sphere can be circumscribed around this polyhedron. | 4.14. Consider some vertex of the polyhedron and three other vertices - the ends of the edges emanating from it. A sphere can be drawn through these 4 points. Such spheres can be constructed for each vertex of the polyhedron, and therefore it is sufficient to prove that for adjacent vertices these spheres coincide.
Le... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,158 |
4.15. Three spheres have a common chord. Through a point of this chord, three chords belonging to different spheres are drawn. Prove that the ends of these three chords lie on one sphere or in one plane. | 4.15. The product of the lengths of the segments into which each of the three chords is divided by the point of their intersection is equal to the product of the lengths of the segments into which the common chord is divided by the point of intersection, which means that all these products are equal. If segments $A B$ ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,159 |
4.16. In space, there are several circles, and any two of them have a pair of common points. Prove that either all these circles have two common points, or they all belong to one sphere (or one plane). | 4.16. If all circles pass through some two points, then everything is proven. Therefore, we can assume that there are three circles, and the third circle does not pass through at least one of the intersection points of the first two circles. We will prove that then these three circles belong to one sphere (or plane). A... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,160 |
4.17. Three circles in space pairwise touch each other (i.e., they have common points and common tangents at these points), and all three points of tangency are distinct. Prove that these circles belong either to the same sphere or to the same plane.
## § 4. Various Problems | 4.17. Let the sphere (or plane) $\alpha$ contain the first and second circles, and the sphere (or plane) $\beta$ contain the second and third circles. Suppose that $\alpha$ and $\beta$ do not coincide. Then their line of intersection is the second circle. Moreover, the common point of the second and third circles also ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,161 |
4.18. Three points $A, B$, and $C$ on a sphere of radius $R$ are connected pairwise (by the shorter) arcs of great circles. Another great circle is drawn through the midpoints of the arcs $A B$ and $A C$, intersecting the extension of the arc $B C$ at point $K$. Find the length of the arc $C K$, if the length of the ar... | 4.18. The plane passing through the center of the sphere and the midpoints of the chords $A B$ and $A C$ also passes through the midpoints of the chords $A B$ and $A C$, and is therefore parallel to the chord $B C$. Consequently, the great circle passing through $B$ and $C$, and the great circle passing through the mid... | \frac{\piR\}{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,162 |
4.19. Chord $A B$ of a sphere with radius 1 has a length of 1 and is positioned at an angle of $60^{\circ}$ to the diameter $C D$ of this sphere. It is known that $A C=\sqrt{2}$ and $A C<B C$. Find the length of the segment $B D$. | 4.19. Let $O$ be the center of the sphere. Take a point $E$ such that $\overrightarrow{C E}=\overrightarrow{A B}$. Since $\angle O C E=60^{\circ}$ and $C E=1=O C$, then $O E=1$. Point $O$ is equidistant from all vertices of the parallelogram $A B E C$, so $A B E C$ is a rectangle and the projection $O_{1}$ of point $O$... | 1 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,163 |
4.20. Given a sphere, a circle on it, and a point $P$ not belonging to the sphere. Prove that the second points of intersection of the sphere with the lines connecting point $P$ to the points of the circle also lie on one circle. | 4.20. Let $A$ and $B$ be two points on a given circle, $A_{1}$ and $B_{1}$ be the second points of intersection of the lines $P A$ and $P B$ with the sphere; $l$ be the tangent to the circumcircle of triangle $P A B$ at point $P$. Then $\angle(l, A P)=\angle(B P, A B)=\angle\left(A_{1} B_{1}, A P\right)$, i.e., $A_{1} ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,164 |
4.21. On a sphere of radius 2, three pairwise tangent circles of radius 1 are arranged. Find the radius of the smallest circle located on the given sphere and tangent to all three given circles. | 4.21. Let $O$ be the center of the sphere; $O_{1}, O_{2}$ and $O_{3}$ be the centers of the given circles; $O_{4}$ be the center of the desired circle. Considering the section of the sphere by the plane $O O_{1} O_{2}$, it is easy to prove that $O O_{1} O_{2}$ is an equilateral triangle with side $\sqrt{3}$. The line $... | 1-\sqrt{2/3} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,165 |
4.22. Let's introduce a coordinate system with the origin $O$ at the center of the Earth, with the axes $O x$ and $O y$ passing through the points on the equator with longitudes $0^{\circ}$ and $90^{\circ}$ respectively, and the axis $\mathrm{Oz}$ passing through the North Pole. What are the coordinates of a point on t... | 4.22. Let $p=(x, y, z)$ be a given point on the Earth's surface, $P^{\prime}$ - the projection on the equatorial plane. Then $z=R \sin \varphi$ and $O P^{\prime}=R \cos \varphi$. Therefore, $x=O P^{\prime} \cos \psi=$ $=R \cos \varphi \cos \psi$ and $y=R \cos \varphi \sin \psi$. Thus, $p=(R \cos \varphi \cos \psi$, $R ... | (R\cos\varphi\cos\psi,R\cos\varphi\sin\psi,R\sin\varphi) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,166 |
4.23. Consider all points on the surface of the Earth for which the geographic latitude equals the longitude. Find the geometric locus of the projections of these points onto the plane of the equator.
## § 5. Area of a Spherical Strip and Volume of a Spherical Segment | 4.23. Let's introduce the same coordinate system as in problem 4.22. If the latitude and longitude of point $P$ are $\varphi$, then $P=$ $=\left(R \cos ^{2} \varphi, R \cos \varphi \sin \varphi, R \sin \varphi\right)$. The projection of this point onto the equatorial plane has coordinates $x=R \cos ^{2} \varphi$ and $y... | (x-R/2)^{2}+y^{2}=R^{2}/4 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,167 |
4.24. Two parallel planes, the distance between which is equal to $h$, intersect a sphere of radius $R$. Prove that the area of the surface of the part of the sphere enclosed between them is $2 \pi R h$. | 4.24. Let's first consider a truncated cone, the lateral surface of which touches a sphere of radius $R$ with center $O$, and the points of tangency divide the generators of the cone in half. We will prove that the area of its lateral surface is $2 \pi R h$, where $h$ is the height of this cone. Let $A B$ be a generato... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,168 |
4.25. Let $A$ be the vertex of a spherical segment, $B$ - a point on the circumference of its base. Prove that the surface area of this segment is equal to the area of a circle with radius $AB$.
保留源文本的换行和格式，直接输出翻译结果如下：
4.25. Let $A$ be the vertex of a spherical segment, $B$ - a point on the circumference of its base.... | 4.25. Let $M$ be the center of the base of the parabolic segment, $h$ the height of the segment, $O$ the center of the sphere, and $R$ the radius of the parabola. Then $A M=h, \quad M O=R-h$ and $B M \perp A O$. Therefore, $A B^{2} \rightarrow$ $-A M^{2}=B M^{2}=B O^{2}-O M^{2}, \quad$ i.e. $A B^{2}=h^{2}+R^{2}-(R-$ - ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,169 |
4.26. Let $h$ be the height of a spherical sector (Fig. 32), $R$ - the radius of the sphere. Prove that the volume of this spherical sector is $2 \pi R^{2} h / 3$. | 4.26. The volume of the steam sector is $S R / 3$, where $S$ is the area of the spherical part of the sector's surface. According to problem 4.24, $S=2 \pi R h$.
| 2\piR^{2}3 | Geometry | proof | Yes | Yes | olympiads | false | 24,170 |
4.27. Let $h$ be the height of the spherical segment (Fig. 33), $R$ be the radius of the sphere. Prove that the volume of the spherical segment is $\pi h^{2}(3 R-h) / 3$. | 4.27. A spherical segment together with the corresponding cone with its vertex at the center of the sphere forms a steam sector. The volume of the spherical sector is $2 \pi R^{2} h / 3$ (problem 4.26). The height of the cone is $R-h$, and the square of the radius of the base is $R^{2} -(R-h)^{2}=2 R h-h^{2}, \quad$ th... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,171 |
4.30. The center of sphere $S_{1}$ lies on sphere $S_{2}$, and these spheres intersect. Prove that the area of the part of the surface of $S_{2}$ that lies inside $S_{1}$ is equal to $1 / 4$ of the surface area of $S_{1}$. | 4.30. Let $O_{1}$ and $O_{2}$ be the centers of spheres $S_{1}$ and $S_{2}$, $R_{1}$ and $R_{2}$ their radii. Let, further, $A$ be the point of intersection of the spheres, $A H$ the height of triangle $O_{1} A O_{2}$. Inside $S_{1}$ is a segment of sphere $S_{2}$ with height $O_{1} H$. Since $\dot{O}_{1} O_{2}=A O_{2}... | \piR_{1}^{2} | Geometry | proof | Yes | Yes | olympiads | false | 24,174 |
4.35. Given a regular tetrahedron with edge length 1. A sphere is tangent to three of its edges, which emanate from one vertex, at their ends. Find the area of the part of the sphere's surface that is located inside the tetrahedron. | 4.35. Consider a regular tetrahedron with edge length 2. The surface of the sphere that touches all its edges is divided by its surface into 4 equal curvilinear triangles, the area of each of which is the desired quantity, and 4 equal segments. Let \( x \) be the distance from the center of a face to a vertex, \( y \) ... | \pi(\frac{1}{\sqrt{3}}-\frac{1}{2}) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,178 |
4.36. On a sphere of radius 2, there are three pairwise tangent circles of radius $\sqrt{2}$. The part of the sphere's surface located outside the circles forms two curvilinear triangles. Find the areas of these triangles.
## § 6. Radical Plane
Let a line $l$, passing through point $O$, intersect the sphere $S$ at po... | 4.36. Consider a cube with edge $2 \sqrt{2}$. A sphere of radius 2 with its center at the center of the cube touches all its edges, and its intersections with the faces are circles of radius $\sqrt{2}$. The surface of the sphere is divided by the surface of the cube into 6 segments and 8 curved triangles. Let $x$ be th... | y=\pi(3\sqrt{2}-4),\quad16\pi-y-3x=\pi(9\sqrt{2}-4) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,179 |
4.37. In space, two non-concentric spheres are given. Prove that the geometric locus of points whose powers relative to these spheres are equal is a plane (the radical plane of the two spheres). | 4.37. Let's introduce a coordinate system with the origin at the center of the first sphere and the $O x$ axis passing through the center of the second sphere. Let the distance between the centers of the spheres be $a$; the radii of the first and second spheres are $R$ and $r$. Then the power of a point $(x, y, z)$ wit... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,180 |
4.38. To two spheres, common tangents $AB$ and $CD$ are drawn. Prove that the lengths of the projections of segments $AC$ and $BD$ onto the line passing through the centers of the spheres are equal. | 4.38. Let $M$ be the midpoint of segment $AB$; $l$ be the line passing through the centers of the given spheres; $P$ be the point of intersection of line $l$ and the radical plane of the given spheres. Since the tangents $MA$ and $MB$, drawn from point $M$ to the given spheres, are equal, $M$ belongs to the radical pla... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,181 |
4.40. Inside a convex polyhedron, there are several non-intersecting pairs of different radii. Prove that this polyhedron can be cut into smaller convex polyhedra, each of which contains exactly one of the given pairs. | 4.40. Let $S_{1}, \ldots, S_{n}$ be the surfaces of given spheres. For each sphere $S_{i}$, consider the figure $M_{i}$, consisting of points whose degree relative to $S_{i}$ is not greater than the degrees relative to all other given spheres. We will prove that the figure $M_{i}$ is convex. Indeed, let $M_{i j}$ be th... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,183 |
4.41. On a sphere, two intersecting circles $S_{1}$ and $S_{2}$ are given. Consider a cone (or cylinder) that is tangent to the given sphere along the circle $S_{1}$. Prove that the circles $S_{1}$ and $S_{2}$ are perpendicular if and only if the plane of the circle $S_{2}$ passes through the vertex of this cone (or is... | 4.41. Let $A$ be the point of intersection of the given circles, $O$ be the vertex of the considered cone (or $OA$ be the generatrix of the cylinder). Since the line $OA$ is perpendicular to the tangent to the circle $S_{1}$ at point $A$, the circles $S_{1}$ and $S_{2}$ are perpendicular if and only if $OA$ is tangent ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,184 |
4.42. Find the area of the curvilinear triangle formed by the intersection of a sphere of radius $R$ with a trihedral angle, the dihedral angles of which are equal to $\alpha, \beta$ and $\gamma$, and the vertex coincides with the center of the sphere. | 4.42. Let us first consider a spherical "bigon," a part of the sphere enclosed within a dihedral angle of magnitude $\alpha$, with its edge passing through the center of the sphere. The area of such a figure is proportional to $\alpha$, and when $\alpha=\pi$, it is equal to $2 \pi R^{2}$; therefore, it is equal to $2 \... | R^{2}(\alpha+\beta+\gamma-\pi) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,185 |
4.43. Let $A_{1}$ and $B_{1}$ be the midpoints of the sides $B C$ and $A C$ of the spherical triangle $A B C$. Prove that the area of the spherical triangle $A_{1} B_{1} C$ is less than half the area of the spherical triangle $A B C$. | 4.43. Consider the set of chords with an endpoint at point $C$, bisected by a larger circle passing through points $A_{1}$ and $B_{1}$. This set forms a circle passing through points $A$ and $B$ and point $C^{\prime}$, which is symmetric to point $C$ with respect to the radius bisecting the arc $A_{1} B_{1}$. The part ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,186 |
4.44. A convex $n$-hedral angle cuts out a spherical $n$-gon on a sphere of radius $R$ with the center at the vertex of the angle. Prove that its area is equal to $R^{2}(\sigma - (n-2) \pi)$, where $\sigma$ is the sum of the dihedral angles. | 4.44. Let's cut an $n$-sided angle into $n-2$ trihedral angles by drawing planes through one of its edges and non-adjacent edges. Writing down the formula from problem 4.42 for each of these trihedral angles and summing them, we obtain the required result. | R^{2}(\sigma-(n-2)\pi) | Geometry | proof | Yes | Yes | olympiads | false | 24,187 |
4.45. Two points $A$ and $B$ are fixed on a sphere. Find the geometric locus of the third vertices $C$ of spherical triangles $A B C$, in which the value $\angle A + \angle B - \angle C$ is constant. | 4.45. Let $M$ and $N$ be the points of intersection of a sphere with a line passing through the center of the circumscribed circle $S$ of triangle $ABC$, perpendicular to its plane. Let $\alpha=\angle MBC=\angle MCB$, $\beta=\angle MAC=\angle MCA$, and $\gamma=\angle MAB=\angle MBA$ (the angles on the sphere are meant)... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,188 |
4.46. Two points $A$ and $B$ are fixed on a sphere. Find the geometric locus of the third vertices $C$ of spherical triangles $A B C$ of a given area. | 4.46. The area of a spherical triangle $ABC$ is determined by the magnitude of $\angle A + \angle B + \angle C$ (see problem 4.42). Let points $A'$ and $B'$ be diametrically opposite to points $A$ and $B$. The angles of the spherical triangles $ABC$ and $A'B'C'$ are related as follows: $\angle A' = \pi - \angle A$ (see... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,189 |
4.47. Three arcs of great circles, each $300^{\circ}$ long, are located on a sphere. Prove that at least two of them have a common point. | 4.47. Suppose that the arcs $a, b$ and $c$ do not intersect. Let $C_{a}$ and $C_{b}$ be the points of intersection of the great circles containing arcs $a$ and $b$. Since arc $a$ is greater than $180^{\circ}$, it contains one of these points, for example $C_{a}$. Then arc $b$ contains point $C_{b}$. Consider also the p... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,190 |
4.48. Several arcs of great circles are given on a sphere, and the sum of their angular magnitudes is less than π. Prove that there exists a plane passing through the center of the sphere and not intersecting any of these arcs.
Consider a sphere of unit radius with its center at the vertex of a polyhedral angle (or on... | 4.48. Let $O$ be the center of the sphere. To each plane passing through $O$, we can associate a pair of points on the sphere - the points of intersection of the perpendicular to this plane passing through the point $O$. It is easy to verify that under this correspondence, planes passing through the point $A$ correspon... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,191 |
4.49. a) Prove that the solid angle of a dihedral angle is equal to $2 \alpha$, where $\alpha$ is the magnitude of the dihedral angle in radians.
b) Prove that the solid angle of a polyhedral angle is equal to $\sigma - (n-2) \pi$, where $\sigma$ is the sum of its dihedral angles. | 4.49. a) The solid angle is proportional to the magnitude of the dihedral angle, and the solid angle of a dihedral angle of magnitude $\pi$ is $2 \pi$.
b) See problem 4.44 | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,192 |
4.50. Calculate the solid angle of a cone with an angle of $2 \alpha$ at the vertex. | 4.50. Let $O$ be the vertex of the cone, $OH$ its height. Construct a sphere of radius 1 with center $O$ and consider its section by a plane passing through the line $OH$. Let $A$ and $B$ be points on the cone lying on the sphere; $M$ be the point of intersection of the ray $OH$ with the sphere (Fig. 40). Then $HM = OM... | 2\pi(1-\cos\alpha) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,193 |
4.51. Prove that the difference between the sum of the solid angles of the dihedral angles of a tetrahedron and the sum of the solid angles of its trihedral angles is \(4 \pi\). | 4.51. The solid angle of a trihedral angle is equal to the sum of its dihedral angles minus $\pi$ (see problem 4.42), therefore, the sum of the solid angles of the trihedral angles of a tetrahedron is equal to twice the sum of its dihedral angles minus $4 \pi$. And twice the sum of the dihedral angles of a tetrahedron ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,194 |
4.52. Prove that the difference between the sum of the solid angles of the dihedral angles at the edges of a polyhedron and the sum of the solid angles of the polyhedral angles at its vertices is $2 \pi(\Gamma-2)$, where $\Gamma-$ is the number of faces of the polyhedron.
## Problems for Independent Solving | 4.52. The solid angle at the $i$-th vertex of a polyhedron is $\sigma_{i}-\left(n_{i}-2\right) \pi$, where $\sigma_{i}$ is the sum of the dihedral angles along the edges emanating from it, and $n_{i}$ is the number of these edges (see problem 4.44). Since each edge emanates from exactly two vertices, $\sum n_{i}=2 \mat... | 2\pi(\Gamma-2) | Geometry | proof | Yes | Yes | olympiads | false | 24,195 |
5.2. Prove that if the dihedral angles of a trihedral angle are right angles, then its plane angles are also right angles. | 5.2. Consider the trihedral angle polar to the given one (see problem 5.1). Its plane angles are right, so its dihedral angles are also right. Consequently, the plane angles of the original trihedral angle are also right. | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,196 |
5.4. Prove that the sum of two plane angles of a trihedral angle is greater than the third plane angle. | 5.4. Consider the trihedral angle $S A B C$ with vertex $S$. The inequality $\angle A S C < \angle A S B$ holds. Then, inside the face $A S C$, we can choose a point $B^{\prime}$ such that $\angle A S B^{\prime} = \angle A S B$ and $S B^{\prime} = S B$, i.e., $\triangle A S B = \triangle A S B^{\prime}$. We can assume ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,198 |
5.5. Prove that the sum of the plane angles of a trihedral angle is less than $2 \pi$, and the sum of its dihedral angles is greater than $\pi$. | 5.5. First solution. On the edges of the trihedral angle, from the vertex $S$, we mark equal segments $S A, S B$, and $S C$. Let $O$ be the projection of point $S$ onto the plane $A B C$. The isosceles triangles $A S B$ and $A O B$ have the same base $A B$ and $A S > A O$. Therefore, $\angle A S B < \pi$.
Second solut... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,199 |
5.6. Ray $S C^{\prime}$ lies inside the trihedral angle $S A B C$ with vertex $S$. Prove that the sum of the plane angles of the trihedral angle $S A B C$ is greater than the sum of the plane angles of the trihedral angle $S A B C^{\prime}$.
## §. Sine and Cosine Theorems for Trihedral Angles | 5.6. Let $K$ be the point of intersection of the line $S C B$ with the line $A C^{\prime}$. According to problem $5.4, \angle C^{\prime} S K + \angle K S B > \angle C^{\prime} S B$ and $\angle C S A + \angle C S K > \angle A S K = \angle A S C^{\prime} + \angle C^{\prime} S K . \quad$ Adding
these inequalities and cons... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,200 |
5.7. Let $\alpha, \beta$ and $\gamma$ be the plane angles of a trihedral angle, and $A, B$ and $C$ be the dihedral angles opposite to them. Prove that $\sin \alpha: \sin A=\sin \beta: \sin B=$ $=\sin \gamma: \sin C$ (the sine theorem for a trihedral angle). | 5.7. Let's take an arbitrary point \( M \) on the edge \( S A \) of the trihedral angle \( S A B C \). Let \( M' \) be the projection of point \( M \) onto the plane \( S B C \), and \( P \) and \( Q \) be the projections of point \( M \) onto the lines \( S B \) and \( S C \). According to the theorem of three perpend... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,201 |
5.8. Let $\alpha, \beta$ and $\gamma$ be the plane angles of a trihedral angle, and $A, B$ and $C$ be the dihedral angles opposite to them.
a) Prove that $\cos \alpha=\cos \beta \cos \gamma+\sin \beta \sin \gamma \cdot \cos A$ (the first cosine theorem for a trihedral angle).
b) Prove that $\cos A=-\cos B \cos C+\sin ... | 5.8. a) The first $p$ e ppe. Take a point $M$ on the edge $S A$ and construct perpendiculars $P M$ and $Q M$ to the edge $S A$ in the planes $S A B$ and $S A C$ (points $P$ and $Q$ lie on the lines $S B$ and $S C$). Expressing the length of side $P Q$ in triangles $P Q M$ and $P Q S$ using the cosine theorem and equati... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,202 |
5.9. The plane angles of a trihedral angle are equal to $\alpha, \beta$ and $\gamma$; the opposite edges form angles $a, b$ and $c$ with the planes of the faces. Prove that $\sin \alpha \sin a=$ $=\sin \beta \sin b=\sin \gamma \sin c$. | 5.9. Let's draw three planes parallel to the faces of a trihedral angle, at a distance of 1 from them, intersecting the edges. Together with the planes of the faces, they form a parallelepiped, with all its heights equal to 1, and therefore, the areas of all its faces are equal. Note now that the lengths of the edges o... | \sin\alpha\sin\sin\beta\sin\sin\gamma\sin | Geometry | proof | Yes | Yes | olympiads | false | 24,203 |
5.10. a) Prove that if all plane angles of a trihedral angle are obtuse, then all its dihedral angles are also obtuse.
b) Prove that if all dihedral angles of a trihedral angle are acute, then all its plane angles are also acute.
## § 4. Various Problems | 5.10. a) According to the first theorem of sines for a trihedral angle (problem $5.8, a$) $\sin \beta \sin \gamma \cos A=\cos \alpha -\cos \beta \cos \gamma$. By the condition $\cos \alpha < 0$, therefore $\mathrm{My} \cos A<0$.
b) For the proof, it is sufficient to use the second cosine theorem (problem 2.8, ). | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,204 |
5.11. Prove that in an arbitrary trihedral angle, the bisectors of two dihedral angles and the angle adjacent to the third dihedral angle lie in the same plane. | 5.11. Pe r oе $р$ eше иo. On the edges of a trihedral angle from the vertex $S$, equal segments $S A, S B$, and $S C$ are laid out. The bisectors of angles $A S B$ and $B S C$ pass through the midpoints of segments $A B$ and $B C$, and the bisector of the angle adjacent to angle $C S A$ is parallel to $C A$.
Second re... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,205 |
5.12. Prove that the pairwise angles between the bisectors of the plane angles of a trihedral angle are either all acute, all obtuse, or all right. | 5.12. Let's lay out vectors a, b, and c of unit length on the edges of a trihedral angle from its vertex. The vectors \(\mathbf{a}+\mathbf{b}, \mathbf{b}+\mathbf{c}\) and \(\mathbf{a}+\mathbf{c}\) define the bisectors of the planar angles. It remains to check that all pairwise scalar products of these vectors have the ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,206 |
5.13. a) A sphere is inscribed in a trihedral angle $SABC$, touching the faces $SBC$, $SCA$, and $SAB$ at points $A_1$, $B_1$, and $C_1$. Express the measure of the angle $ASB_1$ in terms of the plane angles of the given trihedral angle.
b) The inscribed and exscribed spheres of a tetrahedron $ABCD$ touch the face $AB... | 5.13. a) Let $\alpha, \beta$ and $\gamma$ be the plane angles of the trihedral angle $SABC; x=\angle ASB_1=\angle ASC_1, y=\angle BSA_1=\angle BSC_1$ and $z=\angle CSA_1=\angle CSB_1$. Then $x+y=\angle ASC_1+\angle BSC_i=$ $=\angle ASB=\gamma, y+z=\alpha, z+x=\beta$. Therefore, $x=(\beta+$ $+\gamma-\alpha) / 2$.
b) Le... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,207 |
5.14. The plane angles of a trihedral angle are not right angles. Planes are drawn through its edges, perpendicular to the opposite faces. Prove that these planes do not intersect in one line. | 5.14. Let's choose points $A, B$ and $C$ on the edges of a trihedral angle with vertex $S$ such that $S A \perp A B C$ (a plane passing through point $A$ of one edge and perpendicular to it intersects the other two edges, since the planar angles are not right angles). Let $A A_{1}$, $B B_{1}$ and $C C_{1}$ be the altit... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,208 |
5.15. a) The plane angles of a trihedral angle are not right angles. In the planes of its faces, lines are drawn perpendicular to the opposite edges. Prove that all three obtained lines are parallel to one plane.
b) Two trihedral angles with a common vertex $S$ are positioned such that the edges of the second angle li... | 5.15. a) Let a, b, and c be vectors directed along the edges $SA, SB$, and $SC$ of a trihedral angle. The line lying in the plane $SBC$ and perpendicular to the edge $SA$ is parallel to the vector $(\mathbf{a}, \mathbf{b}) \mathbf{c} - (\mathbf{a}, \mathbf{c}) \mathbf{b}$. Similarly, the other two lines are parallel to... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,209 |
5.17. Prove that in a polyhedral angle, any dihedral angle is less than the sum of all the other dihedral angles. | 5.17. Consider the multiray angle $O A_{1} \ldots A_{n}$ with vertex $O$. As follows from the result of problem 5.4, $\angle A_{1} O A_{2} < \angle A_{2} O A_{3} + \angle A_{1} O A_{3}, \quad \angle A_{1} O A_{3} < \angle A_{3} O A_{4} + \angle A_{1} O A_{4}$, $\ldots, \angle A_{1} O A_{n-1} < \angle A_{n-1} O A_{n} + ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,210 |
5.18. One of two convex polyhedral angles with a common vertex lies inside the other. Prove that the sum of the plane angles of the internal polyhedral angle is less than the sum of the plane angles of the external one. | 5.18. Let the polyhedral angle $O A_{\mathrm{i}} \ldots A_{n}$ lie inside the polyhedral angle $O B_{1} \ldots B_{m}$. We can assume that $A_{1}, \ldots, A_{n}$, $B_{i}, \ldots, B_{m}$ are the points of intersection of their edges with a sphere of radius 1. Then the measures of the planar angles of these polyhedral ang... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,211 |
5.19. a) Prove that the sum of the dihedral angles of a convex $n$-hedral angle is greater than $(n-2)\pi$.
b) Prove that the sum of the planar angles of a convex $n$-hedral angle is less than $2\pi$. | 5.19. a) Let's cut the $n$-sided dihedral angle $S A_{1} \ldots A_{n}$ with vertex $S$ into $n-2$ trihedral angles using the planes $S A_{1} A_{3}, S A_{1} A_{4}, \ldots, S A_{1} A_{n-1}$. The sum of the dihedral angles of the $n$-sided dihedral angle is equal to the sum of the dihedral angles of these trihedral angles... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,212 |
5.20. The sum of the planar angles of a certain convex $n$-sided angle is equal to the sum of its dihedral angles. Prove that $n=3$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | 5.20. The sum of the planar angles of an arbitrary convex polyhedral angle is less than $2 \pi$ (see problem 5.19, b), and the sum of the dihedral angles of a convex $n$-hedral angle is greater than $(n-2) \pi$ (see problem 5.19, a). Therefore, $(n-2) \pi < 2\pi$, i.e., $n < 4$. | Algebra | math-word-problem | Yes | Yes | olympiads | false | 24,213 | |
5.21. A sphere is inscribed in a convex dihedral angle of a quadrilateral. Prove that the sums of its opposite planar angles are equal. | 5.21. Let a sphere touch the faces of the tetrahedral angle $S A B C D$ at points $K, L, M$ and $N$ ( $K$ belongs to the face $S A B$, $L$ - to the face $S B C$, etc.). Then $\angle A S K=\angle A S N, \angle B S K=$ $=\angle B S L, \quad \angle C S L=\angle C S M, \quad \angle D S M=\angle D S N . \quad$ Therefore $\a... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,214 |
5.23. On the sides $AB, BC$, and $CA$ of triangle $ABC$ (or on their extensions), points $C_{1}, A_{1}$, and $B_{1}$ are taken.
a) Prove that points $A_{1}, B_{1}$, and $C_{1}$ lie on the same line if and only if
$$
\frac{\overline{A_{1} B}}{\overline{A_{1} C}} \cdot \frac{\overline{B_{1} C}}{\overline{B_{1} A}} \cdot... | 5.23. a) Let when projecting onto a line perpendicular to the line $A_{1} B_{1}$, the points $A, B$, and $C$ go to $A^{\prime}, B^{\prime}$, and $C^{\prime \prime}$, the point $C_{1}$ goes to $Q$, and the two points $A_{1}$ and $B_{1}$ go to the same point $P$. Since $\overline{A_{1} B}: \overline{A_{1} C}=\overline{P ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,216 |
5.24. Given a trihedral angle with vertex $S$ and edges $a, b$, and $c$. The rays $\alpha, \beta$, and $\gamma$ starting from point $S$ are located in the planes of the faces opposite to the edges $a, b$, and $c$ respectively.
a) Prove that the rays $\alpha, \beta$, and $\gamma$ lie in the same plane if and only if
$... | 5.24. a) Let's take arbitrary points \(A, B\), and \(C\) on the edges \(a, b\), and \(c\) of a trihedral angle. Let \(A_{1}, B_{1}\), and \(C_{1}\) be the points where the rays \(\alpha, \beta\), and \(\gamma\) (or their extensions) intersect the lines \(BC, CA\), and \(AB\). Applying the Law of Sines to triangles \(SA... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,217 |
5.25. Given a trihedral angle with vertex $S$ and edges $a, b, c$. Rays $\alpha, \beta$ and $\gamma$ with origin at point $S$ are located in the planes of the faces opposite to the edges $a, b$ and $c$ respectively. We will denote by $l$ the plane containing rays $l$ and $m$.
a) Prove that
$$
\begin{aligned}
& \frac{... | 5.25. a) As in the solution of problem 5.24, a, on the edges $a, b$ and $c$, we can choose points $A, B$ and $C$ such that the rays $\alpha, \beta$ and $\gamma$ are not parallel to the lines $B C, C A$ and $A B$ and intersect them at points $A_{1}, B_{1}$ and $C_{1}$. For brevity, we denote the dihedral angles between ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,218 |
5.26. A sphere is inscribed in a trihedral angle $S A B C$, touching the faces $S B C, S C A$, and $S A B$ at points $A_{1}, B_{1}$, and $C_{1}$ respectively. Prove that the planes $S A A_{1}$, $S B B_{1}$, and $S C C_{1}$ intersect along a single line. | 5.26. Let $a, b$ and $c$ be the edges $SA, SB$ and $SC$; $\alpha, \beta$ and $\gamma$ be the rays $SA_1, SB_1$ and $SC_1$. Since $\angle ASB_1 = \angle ASC_1$, then $|\sin (a, \beta)| = |\sin (a, \gamma)|$. Similarly, $|\sin (b, \alpha)| = |\sin (b, \gamma)|$ and $|\sin (c, \alpha)| = |\sin (c, \beta)|$. Therefore,
$$... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,219 |
5.27. Given a trihedral angle with vertex $S$ and edges $a, b$, and $c$. Rays $\alpha, \beta$, and $\gamma$ are located in the planes of the faces opposite to the edges $a, b$, and $c$, and rays $\alpha^{\prime}, \beta^{\prime}$, and $\gamma^{\prime}$ are symmetric to these rays with respect to the bisectors of the cor... | 5.27. It is easy to check that $\sin (a, \gamma)=-\sin \left(b, \gamma^{\prime}\right)$ and $\sin (b, \gamma)=-\sin \left(a, \gamma^{\prime}\right), \sin (b, \alpha)=-\sin \left(c, \alpha^{\prime}\right)$ and $\sin (c, \alpha)=$ $=-\sin \left(b, \alpha^{\prime}\right), \quad \sin (c, \beta)=-\sin \left(a, \beta^{\prime... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,220 |
5.28. Given a trihedral angle with vertex $S$ and edges $a, b$, and $c$. Lines $\alpha, \beta$, and $\gamma$ are located in the planes of the faces opposite to the edges $a, b$, and $c$. Let $\alpha^{\prime}$ be the line where the plane, symmetric to the plane of $\alpha$ relative to the bisector plane of the dihedral ... | 5.28. Consider the section by a plane passing through the edge a and perpendicular to it, and we will denote the points of intersection of these lines and edges with this plane by the same letters as themselves. There are two cases:
1. The rays $a \alpha$ and $a \alpha^{\prime}$ are symmetric with respect to the bisec... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,221 |
5.29. Given a tetrahedron \( A_{1} A_{2} A_{3} A_{4} \) and some point \( P \). For each edge \( A_{i} A_{j} \), consider the plane that is symmetric to the plane \( P A_{i} A_{j} \) with respect to the bisector plane of the dihedral angle at the edge \( A_{i} A_{j} \). Prove that either all these 6 planes intersect at... | 5.29. Let $\pi_{i j}$ denote the plane symmetric to the plane $P A_{i} A_{j}$ with respect to the bisector plane of the dihedral angle along the edge $A_{i} A_{j}$. As follows from problem 5,28, b, the plane $\pi_{l l}$ passes through the line of intersection of the planes $\boldsymbol{\pi}_{i j}$ and $\pi_{i k}$. Cons... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,222 |
5.30. Given a trihedral angle $S A B C$, where $\angle A S B = \angle A S C = 90^{\circ}$. Planes $\pi_{b}$ and $\pi_{c}$ pass through the edges $S B$ and $S C$, and planes $\pi_{b}^{\prime}$ and $\pi_{c}^{\prime}$ are symmetric to them with respect to the bisector planes of the dihedral angles at these edges. Prove th... | 5.30. The projection onto the plane $B S C$ of any line $l$ passing through the point $S$ coincides with the line at which the plane, passing through the edge $S A$ and the line $l$, intersects the plane $B S C$. Therefore, it is sufficient to prove that the planes, passing through the edge $S A$ and the lines of inter... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,223 |
5.31. Let the Monge point of the tetrahedron \(ABCD\) (see problem 7.32) lie in the plane of the face \(ABC\). Prove that then through the point \(D\) there pass planes in which lie:
a) the points of intersection of the altitudes of the faces \(DAB\), \(DBC\), and \(DAC\)
b) the centers of the circumscribed circles o... | 5.31. a) In solving this problem, we will use the fact that the projection $D_{1}$ of point $D$ onto the plane $A B C$ lies on the circumscribed circle of triangle $A B C$ (problem 7.32, b). We will draw the altitudes $D C_{1}, D A_{1}$, and $D B_{1}$ in triangles $D A B, D B C$, and $D A C$. We need to prove that the ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,224 |
6.1. Do the altitudes of any tetrahedron intersect at one point? | 6.1. No, not in any. Consider a triangle $A B C$ where angle $A$ is not a right angle, and construct a perpendicular $A D$ to the plane of the triangle. In the tetrahedron $A B C D$, the heights drawn from vertices $C$ and $D$ do not intersect. | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,225 |
6.2. a) Through the vertex $A$ of the tetrahedron $ABCD$, three planes are drawn perpendicular to the opposite edges. Prove that all these planes intersect along a single line.
b) Through each vertex of the tetrahedron, a plane is drawn perpendicular to the opposite face and containing the center of the circumscribed ... | 6.2. a) The perpendicular dropped from vertex $A$ to the plane $BCD$ belongs to all three given planes.
b) Check that all the indicated planes pass through the center of the circumscribed sphere of the tetrahedron. | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,226 |
6.3. The median of a tetrahedron is defined as the segment connecting its vertex to the point of intersection of the medians of the opposite face. Express the length of the median of a tetrahedron in terms of the lengths of its edges. | 6.3. Let $A D=a, B D=b, C D=c, B C=a_{1}, C A=b_{1}$ and $A B=c_{1}$. We will compute the length $m$ of the median $D M$. Let $N$ be the midpoint of edge $B C, D N=p$ and $A N=q$. Then $D M^{2}+$ $+M N^{2}-2 D M \cdot M N \cos D M N=D N^{2} \quad$ and $D M^{2}+A M^{2}-$ $-2 D M \cdot A M \cos D M A=A D^{2}$, which mean... | 9^{2}=3(^{2}+b^{2}+^{2})-a_{1}^{2}-b_{1}^{2}-c_{1}^{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,227 |
6.4. Prove that the center of the sphere inscribed in a tetrahedron lies inside the tetrahedron formed by the points of tangency. | 6.4. It is sufficient to prove that if a sphere is inscribed in a trihedral angle, then the plane passing through the points of tangency separates the vertex $S$ of the trihedral angle from the center $O$ of the inscribed sphere. The plane passing through the points of tangency coincides with the plane passing through ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,228 |
6.5. Let $S_{1}$ and $S_{2}$ be the areas of the faces of a tetrahedron adjacent to the edge $a$; $\alpha$ be the dihedral angle at this edge; $b$ be the edge opposite to $a$; $\varphi$ be the angle between the edges $b$ and $a$. Prove that
$$
S_{1}^{2}+S_{2}^{2}-2 S_{1} S_{2} \cos \alpha=(a b \sin \varphi)^{2} / 4
$$ | 6.5. The projection of a tetrahedron onto a plane perpendicular to edge $a$ is a triangle with sides $2 S_{1} / a, 2 S_{2} / a$ and $b \sin \varphi ;$ the angle between the first two sides is $\alpha$. Writing the cosine theorem for this triangle, we obtain the required, | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,229 |
6.6. Prove that the product of the lengths of two opposite edges of a tetrahedron, divided by the product of the cosines of the dihedral angles at these edges, is the same for all three pairs of opposite edges of the tetrahedron (the cosine theorem for a tetrahedron). | 6.6. Consider the tetrahedron $A B C D$. Let $A B = a$, $C D = b$; $\alpha$ and $\beta$ - the dihedral angles at the edges $A B$ and $C D$; $S_{1}$ and $S_{2}$ - the areas of the faces $A B C$ and $A B D$, $S_{3}$ and $S_{4}$ - the areas of the faces $C D A$ and $C D B$; $V$ - the volume of the tetrahedron. According t... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,230 |
6.7. a) Let $S_{1}, S_{2}, S_{3}$ and $S_{4}$ be the areas of the faces of a tetrahedron; $P_{1}, P_{2}$ and $P_{3}$ be the areas of the faces of a parallelepiped, the faces of which pass through the edges of the tetrahedron parallel to its opposite edges. Prove that,
$$
S_{1}^{2}+S_{2}^{2}+S_{3}^{2}+S_{4}^{2}=P_{1}^{... | 6.7. a) Let $\alpha, \beta$ and $\gamma$ be the dihedral angles at the edges of the face with area $S_{1}$. Then $S_{1}=S_{2} \cos \alpha+S_{3} \cos \beta+S_{4} \cos \gamma$ (see problem 2.13). Moreover, according to problem 6.5,
\[
\begin{aligned}
& S_{1}^{2}+S_{2}^{2}-2 S_{1} S_{2} \cos \alpha=P_{1}^{2} \\
& S_{1}^{... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,231 |
6.8. Let $S_{i}, R_{i}$ and $l_{i} (i=1,2,3,4)$ be the areas of the faces, the radii of the circles circumscribed around these faces, and the distances from the centers of these circles to the opposite vertices of the tetrahedron. Prove that $18 V^{2}=$ $=\sum_{i=1}^{4} S_{i}^{2}\left(l_{i}^{2}-R_{i}^{2}\right)$, where... | 6.8. We will first prove the case when the center of the circumscribed sphere is inside the tetrahedron. First, we will prove that \( l_{i}^{2} - R_{i}^{2} = 2 h_{i} d_{i} \), where \( d_{i} \) is the distance from the center of the circumscribed sphere to the \( i \)-th face, \( h_{i} \) is the height of the tetrahedr... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,232 |
6.9. Prove that for any tetrahedron, there exists a triangle whose side lengths are equal to the products of the lengths of opposite edges of the tetrahedron, and that the area \( S \) of this triangle is equal to \( 6 V R \), where \( V \) is the volume of the tetrahedron, and \( R \) is the radius of its circumscribe... | 6.9. Let the lengths of the edges $AD, BD$ and $CD$ be $a, b$ and $c$; the lengths of the edges $BC, CA$ and $AB$ be $a', b'$ and $c'$. Draw a plane П through vertex $D$ tangent to the circumscribed sphere of the tetrahedron. Consider the tetrahedron $A_1BC_1D$, formed by the planes П, $BCD, ABD$ and the plane passing ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,233 |
6.10. Let $a$ and $b$ be the lengths of two skew edges of a tetrahedron, $\alpha$ and $\beta$ - the dihedral angles at these edges. Prove that the value $a^{2}+b^{2}+2 a b \operatorname{ctg} \alpha \times$ $\times \operatorname{ctg} \beta$ does not depend on the choice of the pair of skew edges (Bretschneider's theorem... | 6.10. Let $S_{1}$ and $S_{2}$ be the areas of the faces sharing a common edge $a$, and $S_{3}$ and $S_{i}$ be the areas of the faces sharing a common edge $b$. Further, let $a$, $m$, and $n$ be the lengths of the edges of the face with area $S_{1}$; $\alpha, v$, and $\delta$ be the dihedral angles at these edges; $h_{1... | ^{2}+b^{2}+2\operatorname{ctg}\alpha\operatorname{ctg}\beta=(2Q-T)/9V^{2} | Geometry | proof | Yes | Yes | olympiads | false | 24,234 |
6.11. Prove that for any tetrahedron, there exist no fewer than 5 and no more than 8 spheres, each of which touches all the planes of its faces.
## § 2. Tetrahedra Possessing Special Properties | 6.11. Let $V$ be the volume of a tetrahedron; $S_{1}, S_{2}, S_{3}$, and $S_{4}$ be the areas of its faces. If the distance from point $O$ to the $i$-th face is $h_{i}$, then $\left(\sum \varepsilon_{i} h_{i} S_{i}\right) / 3=V$, where $\varepsilon_{i}=+1$ if point $O$ and the tetrahedron lie on the same side of the $i... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,235 |
6.12. In a triangular pyramid $S A B C$ with vertex $S$, the lateral edges are equal, and the sum of the dihedral angles at edges $S A$ and $S C$ is $180^{\circ}$. Express the length of the lateral edge in terms of the sides $a$ and $c$ of triangle $A B C$. | 6.12. Let's take a point \( A_{1} \) on the ray \( A S \) such that \( A A_{1} = 2 A S \). In the pyramid \( S A_{1} B C \), the dihedral angles at the edges \( S A_{1} \) and \( S C \) are equal, and \( S A_{1} = S C \), so \( A_{1} B = C B = a \). The triangle \( A B A_{1} \) is a right triangle because its median \(... | \frac{\sqrt{^2+^2}}{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,236 |
6.13. The sum of the lengths of one pair of opposite edges of a tetrahedron is equal to the sum of the lengths of the other pair. Prove that the sum of the dihedral angles at the first pair of edges is equal to the sum of the dihedral angles at the second pair. | 6.13. If in the tetrahedron \(ABCD\) the sum of the lengths of the edges \(AB\) and \(CD\) is equal to the sum of the lengths of the edges \(BC\) and \(AD\), then there exists a sphere that touches these four edges at internal points (see problem 8.30). Let \(O\) be the center of this sphere. Note now that if tangents ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,237 |
6.14. All faces of the tetrahedron are similar right-angled triangles. Find the ratio of the greatest edge to the smallest. | 6.14. Let $a$ be the length of the longest edge of the tetrahedron. In both faces adjacent to this edge, it is the hypotenuse. These faces are equal, as similar right triangles with a common hypotenuse are equal; let $m$ and $n$ be the lengths of the legs of these right triangles, and $b$ be the length of the sixth edg... | \sqrt{\frac{1+\sqrt{5}}{2}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,238 |
6.15. The edge of a regular tetrahedron $ABCD$ is equal to $a$. The vertices of the spatial quadrilateral $A_{1} B_{1} C_{1} D_{1}$ lie on the corresponding faces of the tetrahedron ($A_{1}$ - on the face opposite to $A$, and so on), and its sides are perpendicular to the faces of the tetrahedron: $A_{1} B_{1} \perp BC... | 6.15. Drop perpendiculars $A_{1} K$ and $B_{1} K$ to $C D$, $B_{1} L$ and $C_{1} L$ to $A D$, $C_{1} M$ and $D_{1} M$ to $A B$, $D_{1} N$ and $A_{1} N$ to $B C$. The lengths of these perpendiculars are equal to the cosine of the dihedral angle at the edge of a regular tetrahedron, i.e., they are equal to $1 / 3$ (see p... | \frac{}{\sqrt{6}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,239 |
6.16. A sphere touches the edges $A B, B C, C D$ and $D A$ of the tetrahedron $A B C D$ at points $L, M, N$ and $K$, which are vertices of a square. Prove that if this sphere touches the edge $A C$, then it also touches the edge $B D$.
| 6.16. But the quadrilateral $K L M N$ is a square. We draw planes through the points $K, L, M$, and $N$, tangent to the sphere. Since all these planes are equally inclined to the plane $K L M N$, they intersect at one point $S$, located on the line $O O_{1}$, where $O$ is the center of the sphere, and $O_{1}$ is the ce... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,240 |
6.17. Let $M$ be the center of mass of the tetrahedron $ABCD$, and $O$ be the center of its circumscribed sphere.
a) Prove that the lines $DM$ and $OM$ are perpendicular if and only if $AB^2 + BC^2 + CA^2 = AD^2 + BD^2 + CD^2$.
b) Prove that if the points $D$ and $M$ and the points of intersection of the medians of t... | 6.17. a) Let $BC=a, CA=b, AB=c, DA=a_{1}, DB=b_{1}$ and $DC=c_{1}$. Let, further, $G$ be the point of intersection of the medians of triangle $ABC$, $N$ be the point of intersection of the line $DM$ with the circumscribed sphere, $K$ be the point of intersection of the line $AG$ with the circumscribed circle of triangl... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,241 |
6.18. In the tetrahedron $A B C D$, the plane angles at vertex $D$ are right angles. Let $\angle C A D=\alpha, \angle C B D=\beta$, and $\angle A C B=\varphi$. Prove that $\cos \varphi=\sin \alpha \sin \beta$. | 6.18. Let $CD = a$. Then $AC = a / \sin \alpha, BC = a / \sin \beta$ and $AB = a \sqrt{\operatorname{ctg}^{2} \alpha + \operatorname{ctg}^{2} \beta}$. Considering that $AB^{2} = AC^{2} + BC^{2} - 2 \cdot AC \cdot BC \cos \varphi$, we obtain the required result. | \cos\varphi=\sin\alpha\sin\beta | Geometry | proof | Yes | Yes | olympiads | false | 24,242 |
6.19. All plane angles at one vertex of the tetrahedron are right angles. Prove that the lengths of the segments connecting the midpoints of its opposite edges are equal. | 6.19. Consider a rectangular parallelepiped with edges $A B, A D$ and $A A_{1}$ which are edges of the given tetrahedron. The segment connecting the midpoints of edges $A B$ and $A_{1} D$ is the midline of triangle $A B D_{1}$ (parallel to $B D_{1}$); therefore, its length is $d / 2$, where $d$ is the length of the dia... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,243 |
6.20. In the tetrahedron \(ABCD\), the plane angles at vertex \(D\) are right angles. Let \(h\) be the height of the tetrahedron dropped from vertex \(D\); \(a, b\), and \(c\) be the lengths of the edges emanating from vertex \(D\). Prove that
\[
\frac{1}{h^{2}}=\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}
\] | 6.20. Since $S_{A B C}^{2}=S_{A B D}^{2}+S_{B C D}^{2}+S_{A C D}^{2}$ (see problem 1.22), then $S_{A B C}=\sqrt{a^{2} b^{2}+b^{2} c^{2}+a^{2} c^{2}} / 2$. Therefore, the volume of the tetrahedron is $h \sqrt{a^{2} b^{2}+b^{2} c^{2}+a^{2} c^{2}} /$. On the other hand, it is equal to $a b c / 6$. By equating these expres... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,244 |
6.21. In the tetrahedron \(ABCD\), the plane angles at vertex \(A\) are right angles, and \(AB = AC + AD\). Prove that the sum of the plane angles at vertex \(B\) is \(90^\circ\). | 6.21. Let's take points $P$ and $R$ on rays $A C$ and $A D$ such that $A P = A R = A B$, and consider the square $A P Q R$. It is clear that $\triangle A B C = \triangle R Q D$ and $\triangle A B D = \triangle P Q C$, and therefore, $\triangle B C D = \triangle Q D C$. Thus, the sum of the planar angles at vertex $B$ i... | 90 | Geometry | proof | Yes | Yes | olympiads | false | 24,245 |
6.22. Three dihedral angles of a tetrahedron are right angles. Prove that this tetrahedron has three right planar angles. | 6.22. For each edge of the tetrahedron, there exists only one edge that is not adjacent to it, but among any three edges, there will be two that are adjacent. Note now that there cannot be three dihedral angles at the edges of one face. Therefore, there are two possible configurations for the three edges, the dihedral ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,246 |
6.24. Three dihedral angles of a tetrahedron, not belonging to the same vertex, are equal to $90^{\circ}$, and all other dihedral angles are equal to each other. Find these angles.
## § 4. Equifacial Tetrahedron
Definition. A tetrahedron is called equifacial if all its faces are equal, i.e., its opposite edges are pa... | 6.24. As follows from the solution of problem 6.22, the vertices of the given tetrahedron are the vertices $A, B, D$ and $D_{1}$ of the rectangular parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$. Let $\alpha$ be the required angle; $A B=a, A D=b$ and $D D_{1}=c$. Then $a=$ $=b \operatorname{tg} \alpha$ and $c=b \oper... | \alpha=\arccos(\frac{\sqrt{5}-1}{2}) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 24,247 |
6.25. Prove that all faces of a tetrahedron are equal if and only if one of the following conditions is satisfied:
a) the sum of the plane angles at some vertex is $180^{\circ}$ and, in addition, there are two pairs of equal opposite edges in the tetrahedron;
b) the centers of the inscribed and circumscribed spheres ... | 6.25. a) Let $AB = CD$, $AC = BD$, and the sum of the planar angles at vertex $A$ is $180^{\circ}$. To prove that $AD = BC$, it is sufficient to check that $\angle ACD = \angle BAC$. Since the sum of the angles in triangle $ACD$ and the sum of the planar angles at vertex $A$ are both $180^{\circ}$, and $\angle DAB = \a... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,248 |
6.26. In the tetrahedron $A B C D$, the dihedral angles at the edges $A B$ and $D C$ are equal; the dihedral angles at the edges $B C$ and $A D$ are also equal. Prove that $A B=D C$ and $B C=A D$. | 6.26. The trihedral angles at vertices $A$ and $C$ have equal dihedral angles, so they are equal (problem 5.3). Therefore, their plane angles are equal, which means $\triangle A B C=\triangle C D A$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,249 |
6.27. A line passing through the center of mass of a tetrahedron and the center of its circumscribed sphere intersects the edges $A B$ and $C D$. Prove that $A C = B D$ and $A D = B C$. | 6.27. The center of mass of the tetrahedron lies on the line connecting the midpoints of edges \(AB\) and \(CD\). Therefore, on this line lies
Fig. 48 the center of the circumscribed sphere... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,250 |
6.28. A line passing through the center of mass of a tetrahedron and the center of its inscribed sphere intersects the edges $A B$ and $C D$. Prove that $A C = B D$ and $A D = B C$. | 6.28. Let $K$ and $L$ be the midpoints of the edges $AB$ and $CD$. The center of mass of the tetrahedron lies on the line $KL$, so the center of the inscribed sphere also lies on the line $KL$. Consequently, when projecting onto a plane perpendicular to $CD$, the segment $KL$ transforms into the bisector of the triangl... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,251 |
6.29. Prove that if $\angle B A C=\angle A B D=$ $=\angle A C D=\angle B D C$, then the tetrahedron $A B C D$ is isosceles. | 6.29. Let $S$ be the midpoint of edge $B C ; K, L, M$ and $N$ be the midpoints of edges $A B, A C, D C$ and $D B$. Then $S K L M N$ is a tetrahedral angle with equal planar angles, and its section $K L M N$ is a parallelogram. On the one hand, a tetrahedral angle with equal planar angles has a section - a rhombus (prob... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,252 |
6.30. Given a tetrahedron \(ABCD\); \(O_{e}\), \(O_{k}\), \(O_{c}\), and \(O_{d}\) are the centers of the exinscribed spheres, touching its faces \(BCD\), \(ACD\), \(ABD\), and \(ABC\). Prove that if the trihedral angles \(O_{a} BCD\), \(O_{s} ACD\), \(O_{c} ABD\), and \(O_{d} ABC\) are right angles, then all faces of ... | 6.30. The point of tangency of the exsphere with the face $A B C$ coincides with the projection $H$ of the point $O_{d}$ (the center of the sphere) onto the plane $A B C$. Since the trihedral angle $O_{d} A B C$ is right, $H$ is the point of intersection of the altitudes of the triangle $A B C$ (see problem 2.11).
Let... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,253 |
6.33. In a regular tetrahedron \(ABCD\), the height \(AH\) is dropped; \(H_1\) is the point of intersection of the heights of the face \(BCD\); \(h_1\) and \(h_2\) are the lengths of the segments into which one of the heights of the face \(BCD\) is divided by the point \(H_1\).
a) Prove that the points \(H\) and \(H_1... | 6.33. Complete the given tetrahedron to a rectangular parallelepiped. Let \( A A_{1} \) be its diagonal, \( O \) be its center. The point \( H_{i} \) is the projection of the point \( A_{i} \) onto the face \( B C D \) (see problem 2.11), and the center \( O_{1} \) of the circumscribed circle of triangle \( B C D \) is... | proof | Geometry | proof | Yes | Yes | olympiads | false | 24,255 |
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