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equal angles with a sphere inscribed into a prism - sharygin

Source: Sharygin Geometry Olympiad 2016 First Round P24 grade 11

July 25, 2018
geometry3D geometrysphereprism

Problem Statement

A sphere is inscribed into a prism ABCA′B′C′ABCA'B'C' and touches its lateral faces BCC′B′,CAA′C′,ABB′A′BCC'B', CAA'C', ABB'A' at points Ao,Bo,CoA_o, B_o, C_o respectively. It is known that ∠AoBB′=∠BoCC′=∠CoAA′\angle A_oBB' = \angle B_oCC' =\angle C_oAA'. a) Find all possible values of these angles. b) Prove that segments AAo,BBo,CCoAA_o, BB_o, CC_o concur. c) Prove that the projections of the incenter to A′B′,B′C′,C′A′A'B', B'C', C'A' are the vertices of a regular triangle.
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