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PQR Intersections

Source: KMO 2021 P1

November 13, 2021
geometryangle bisectorcircumcircle

Problem Statement

Let ABCABC be an acute triangle and DD be an intersection of the angle bisector of AA and side BCBC. Let Ω\Omega be a circle tangent to the circumcircle of triangle ABCABC and side BCBC at AA and DD, respectively. Ω\Omega meets the sides AB,ACAB, AC again at E,FE, F, respectively. The perpendicular line to ADAD, passing through E,FE, F meets Ω\Omega again at G,HG, H, respectively. Suppose that AEAE and GDGD meet at PP, EHEH and GFGF meet at QQ, and HDHD and AFAF meet at RR. Prove that QF‾QG‾=HR‾PG‾\dfrac{\overline{QF}}{\overline{QG}}=\dfrac{\overline{HR}}{\overline{PG}}.
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