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O coincides the incenter

Source: Iranian third round 2015 geometry problem 5

September 10, 2015
geometrycircumcircleincenter

Problem Statement

Let ABCABC be a triangle with orthocenter HH and circumcenter OO. Let RR be the radius of circumcircle of △ABC\triangle ABC. Let A′,B′,C′A',B',C' be the points on AH→,BH→,CH→\overrightarrow{AH},\overrightarrow{BH},\overrightarrow{CH} respectively such that AH.AA′=R2,BH.BB′=R2,CH.CC′=R2AH.AA'=R^2,BH.BB'=R^2,CH.CC'=R^2. Prove that OO is incenter of △A′B′C′\triangle A'B'C'.
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