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BF = CG wanted, BD = EC given, line // angle bisector

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2018 Shortlist G2

October 2, 2021
geometryequal segmentsangle bisector

Problem Statement

Consider △ABC\vartriangle ABC, with ADAD bisecting ∠BAC\angle BAC, DD on segment BCBC. Let EE be a point on BCBC, such that BD=ECBD = EC. Through EE we draw the line ℓ\ell parallel to ADAD and consider a point P P on it and inside the △ABC\vartriangle ABC. Let GG be the point where line BPBP cuts side ACAC and let F be the point where line CPCP to side ABAB. Show that BF=CGBF = CG.
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