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Concyclic incentres and excentres

Source: Indian Team Selection Test 2015 Day 1 Problem 1

July 11, 2015
geometry

Problem Statement

Let ABCDABCD be a convex quadrilateral and let the diagonals ACAC and BDBD intersect at OO. Let I1,I2,I3,I4I_1, I_2, I_3, I_4 be respectively the incentres of triangles AOB,BOC,COD,DOAAOB, BOC, COD, DOA. Let J1,J2,J3,J4J_1, J_2, J_3, J_4 be respectively the excentres of triangles AOB,BOC,COD,DOAAOB, BOC, COD, DOA opposite OO. Show that I1,I2,I3,I4I_1, I_2, I_3, I_4 lie on a circle if and only if J1,J2,J3,J4J_1, J_2, J_3, J_4 lie on a circle.
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