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intersection of circumcircles lies on diagonal of ABCD, equal angles given

Source: 239 MO 2001 VIII-IX p2

May 7, 2020
geometrycircumcirclediagonalequal angles

Problem Statement

In a convex quadrangle ABCD ABCD , the rays DA DA and CB CB intersect at point Q Q , and the rays BA BA and CD CD at the point P P . It turned out that ∠AQB=∠APD \angle AQB = \angle APD . The bisectors of the angles ∠AQB \angle AQB and ∠APD \angle APD intersect the sides quadrangle at points X X , Y Y and Z Z , T T respectively. Circumscribed circles of triangles ZQT ZQT and XPY XPY intersect at K K inside quadrangle. Prove that K K lies on the diagonal AC AC .
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