By V. Gusev, V. Litvinenko, A. Morkovich
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Additional resources for Problemas Matematicos - Geometria
For tetrahedrons cut off by planes A1 KD and A1 LB the tetrahedron A1 BP Q 1 whose volume is equal to 24 V is a common one. Therefore, the volume of the common part of tetrahedrons ABCD and A1 B1 C1 D1 is equal to V (1 − 3 3V 3 + )= . 4 24 8 SOLUTIONS 37 Figure 31 (Sol. 32. The ratio of the segments of the edge is equal to the ratio of the heights dropped from its endpoints to the bisector plane and the latter ratio is equal to the ratio of volumes of tetrahedrons into which the bisector plane divides the given tetrahedron.
Let us consider the section of given spheres by planes passing through points O1 SOLUTIONS 53 Figure 37 (Sol. 39) and O2 and draw outer and inner tangents to the circles obtained in the section (Fig. 37). Let P and Q be the midpoints of these tangents. Let us prove that N Q ≤ N M ≤ N P . 4). Thus the locus to be found is the annulus situated in the radical plane; the outer radius of the annulus is N P and the inner one is N Q. 40. Let S1 , . . , Sn be the surfaces of the given balls. For every sphere Si consider figure Mi that consists of points whose degree with respect to Si does not exceed the degrees relative to all the other spheres.
In what ratio the volume of the cube is divided by the plane passing through one of the cubes vertices and the centers of the two faces that do not contain this vertex? 42. Segment EF does not lie in plane of the parallelogram ABCD. Prove that the volume of tetrahedron EF AD is equal to either sum or difference of volumes of tetrahedrons EF AB and EF AC. 28 CHAPTER 3. 43. The lateral faces of an n-gonal pyramid are lateral faces of regular quadrangular pyramids. The vertices of the bases of quadrangular pyramids distinct from the vertices of an n-gonal pyramid pairwise coincide.