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insphere of pyramid, planar cross-sections with equal area

Source: Bulgaria 1985 P3

June 16, 2021
geometry3D geometrypyramidarea

Problem Statement

A pyramid MABCDMABCD with the top-vertex MM is circumscribed about a sphere with center OO so that OO lies on the altitude of the pyramid. Each of the planes ACM,BDM,ABOACM,BDM,ABO divides the lateral surface of the pyramid into two parts of equal areas. The areas of the sections of the planes ACMACM and ABOABO inside the pyramid are in ratio (2+2):4(\sqrt2+2):4. Determine the angle δ\delta between the planes ACMACM and ABOABO, and the dihedral angle of the pyramid at the edge ABAB.
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