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Cute Geo

Source: Korea National Olympiad 2020 P6

November 25, 2020
geometryKoreapentagon

Problem Statement

Let ABCDEABCDE be a convex pentagon such that quadrilateral ABDEABDE is a parallelogram and quadrilateral BCDEBCDE is inscribed in a circle. The circle with center CC and radius CDCD intersects the line BD,DEBD, DE at points F,G(≠D)F, G(\neq D), and points A,F,GA, F, G is on line l. Let HH be the intersection point of line ll and segment BCBC. Consider the set of circle Ω\Omega satisfying the following condition.
Circle Ω\Omega passes through A,HA, H and intersects the sides AB,AEAB, AE at point other than AA.
Let P,Q(≠A)P, Q(\neq A) be the intersection point of circle Ω\Omega and sides AB,AEAB, AE. Prove that AP+AQAP+AQ is constant.
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