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Additional resources for 3-D Shapes Are Like Green Grapes!
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 equal to 4π. 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 equal to 2π(F − 2), where F is the number of faces of the polyhedron. 53. Through point D, three lines intersecting a sphere at points A and A1 , B and B1 , C and C1 , respectively, are drawn.
The vertices of the bases of quadrangular pyramids distinct from the vertices of an n-gonal pyramid pairwise coincide. Find the ratio of volumes of the pyramids. 44. The dihedral angle at edge AB of tetrahedron ABCD is a right one; M is the midpoint of edge CD. Prove that the area of triangle AM B is a half area of the parallelogram whose diagonals are equal to and parallel to edges AB and CD. 45. Faces ABD, BCD and CAD of tetrahedron ABCD serve as lower bases of the three prisms; the planes of their upper bases intersect at point P .
15. The product of the lengths of segments into which the intersection point divides each of the chords is equal to the product of the lengths of segments into which the common chord is divided by their intersection point, hence, these products are equal. If segments AB and CD intersect at point O and AO · OB = CO · OD, then points A, B, C and D lie on one circle. Therefore, the endpoints of the first and second chords, as well as the endpoints of the second and third chords, lie on one circle.