SHARYGIN 2016 Problem 24

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