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Turkey NMO 2017 p2

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January 25, 2018
geometry

Problem Statement

Let ABCDABCD be a quadrilateral such that line ABAB intersects CDCD at XX. Denote circles with inradius r1r_1 and centers A,BA, B as waw_a and wbw_b with inradius r2r_2 and centers C,DC, D as wcw_c and wdw_d. waw_a intersects wdw_d at P,QP, Q. wbw_b intersects wcw_c at R,SR, S. Prove that if XA.XB+r22=XC.XD+r12XA.XB+r_2^2=XC.XD+r_1^2, then P,Q,R,SP,Q,R,S are cyclic.
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