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By Prasolov V.V., Sharygin I.F.
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Extra resources for Problems in plane and solid geometry. Solid geometry
Example text
Prove that all these spheres have a common tangent line. 7. A sphere with diameter CE is tangent to plane ABC at point C; line AD is tangent to the sphere. Prove that if point B lies on line DE, then AC = AB. 8. Given cube ABCDA1 B1 C1 D1 . A plane passing through vertex A and tangent to the sphere inscribed in the cube intersects edges A1 B1 and A1 D1 at points K and N , respectively. Find the value of the angle between planes AC1 K and AC1 N . 9. Two equal triangles KLM and KLN have a common side KL, moreover, ∠KLM = ∠LKN = 60◦ , KL = 1 and LM = KN = 6.
A) Two circles not in one plane intersect at two distinct points, A and B. Prove that there exists a unique sphere that contains these circles. Typeset by AMS-TEX 40 CHAPTER 4. SPHERES b) Two circles not in one plane are tangent to line l at point P . Prove that there exists a unique sphere containing these circles. 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. 14. All the faces of a convex polyhedron are inscribed polygons and all the angles are trihedral ones.
28) then subtract the volume of three pyramids similar to the initial one with coefficient 5 9 and add the volume of three pyramids similar to the initial one with coefficient 1 9 . Therefore, the volume of the common part is equal to 1 5 1 110V V (1 − ( )3 − 3( )3 + 3( )3 ) = . 3 9 9 243 d) The common part is depicted on Fig. 30 d). Its volume is equal to 7 1 12V 3 . 29. 26). 2 it is easy to prove that V3 =( abc 3 2 ) p q, 6 where a, b and c are the lengths of the edges coming out of vertex A; p the product of the sines of the plane angles at vertex A; q the product of the sines of dihedral 36 CHAPTER 3.