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Geometry Mathley 14.3 <DGE + <FGQ = 180^o

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June 7, 2020
incirclegeometryanglesequal segments

Problem Statement

Let ABCABC be a triangle inscribed in circle (I)(I) that is tangent to the sides BC,CA,ABBC,CA,AB at points D,E,FD,E, F respectively. Assume that LL is the intersection of BEBE and CF,GCF,G is the centroid of triangle DEF,KDEF,K is the symmetric point of LL about GG. If DKDK meets EFEF at P,QP, Q is on EFEF such that QF=PEQF = PE, prove that ∠DGE+∠FGQ=180o\angle DGE + \angle FGQ = 180^o.
Nguyễn Minh Hà
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