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Junior Balkan Mathematical Olympiad 2024- P2

Source: JBMO 2024

June 27, 2024
geometrycircumcirclecollinearBalkanJBMO

Problem Statement

Let ABCABC be a triangle such that AB<ACAB < AC. Let the excircle opposite to A be tangent to the lines AB,ACAB, AC, and BCBC at points D,ED, E, and FF, respectively, and let JJ be its centre. Let PP be a point on the side BCBC. The circumcircles of the triangles BDPBDP and CEPCEP intersect for the second time at QQ. Let RR be the foot of the perpendicular from AA to the line FJFJ. Prove that the points P,QP, Q, and RR are collinear.
(The excircle of a triangle ABCABC opposite to AA is the circle that is tangent to the line segment BCBC, to the ray ABAB beyond BB, and to the ray ACAC beyond CC.)
Proposed by Bozhidar Dimitrov, Bulgaria
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