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Part of 2018 Rioplatense Mathematical Olympiad, Level 3

Problems(1)

equal segments related to 3 circumcircles

Source: Rioplatense Olympiad 2018 level 3 p2

12/11/2018
Let PP be a point outside a circumference Γ\Gamma, and let PAPA be one of the tangents from PP to Γ\Gamma. Line ll passes through PP and intersects Γ\Gamma at BB and CC, with BB between PP and CC. Let DD be the symmetric of BB with respect to PP. Let ω1\omega_1 and ω2\omega_2 be the circles circumscribed to the triangles DACDAC and PABPAB respectively. ω1\omega_1 and ω2\omega _2 intersect at E≠AE \neq A. Line EBEB cuts back to ω1\omega _1 in FF. Prove that CF=ABCF = AB.
geometrycircumcircleequal segmentssymmetry