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Convex quadrilaterals and equal ratios

Source: Netherlands IMO Team Selection Test 2013

October 21, 2014
ratiogeometrycircumcirclegeometry unsolved

Problem Statement

Let PP be the point of intersection of the diagonals of a convex quadrilateral ABCDABCD.Let X,Y,ZX,Y,Z be points on the interior of AB,BC,CDAB,BC,CD respectively such that AXXB=BYYC=CZZD=2\frac{AX}{XB}=\frac{BY}{YC}=\frac{CZ}{ZD}=2. Suppose that XYXY is tangent to the circumcircle of △CYZ\triangle CYZ and that YZY Z is tangent to the circumcircle of △BXY\triangle BXY.Show that ∠APD=∠XYZ\angle APD=\angle XYZ.
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