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X_1X_2/X_3X_4=Y_1Y_2/Y_3Y_4 if X_1,X_2,Y_1,Y_2 concyclic, excenter related

Source: 2020 XXIII All-Ukrainian Tournament of Young Mathematicians named after M. Y. Yadrenko, Qualifying p10

May 4, 2021
excentergeometryratioConcyclicUkrainian TYM

Problem Statement

In triangle ABCABC, point II is the center, point IaI_a is the center of the excircle tangent to the side BCBC. From the vertex AA inside the angle BACBAC draw rays AXAX and AYAY. The ray AXAX intersects the lines BIBI, CICI, BIaBI_a, CIaCI_a at points X1X_1, ......, X4X_4, respectively, and the ray AYAY intersects the same lines at points Y1Y_1, ......, Y4Y_4 respectively. It turned out that the points X1,X2,Y1,Y2X_1,X_2,Y_1,Y_2 lie on the same circle. Prove the equality X1X2X3X4=Y1Y2Y3Y4.\frac{X_1X_2}{X_3X_4}= \frac{Y_1Y_2}{Y_3Y_4}.
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