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In a right ∆ABC prove that A,Q,R are collinear & AP=AC

Source: JBMO Shortlist 2010 Problem G1 (Geometry #1)

January 25, 2015
geometrycircumcirclereflection

Problem Statement

<spanclass=′latex−bold′>ProblemG1</span><span class='latex-bold'>Problem G1</span> Consider a triangle ABCABC with ∠ACB=90∘\angle ACB=90^{\circ}. Let FF be the foot of the altitude from CC. Circle ω\omega touches the line segment FBFB at point PP, the altitude CFCF at point QQ and the circumcircle of ABCABC at point RR. Prove that points A,Q,RA, Q, R are collinear and AP=ACAP = AC.
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