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circle with diameter a side of a triangle

Source: Nordic Mathematical Contest 2015 #1

September 23, 2017

geometry

Problem Statement

Let

{ABC}

be a triangle and

{\Gamma}

the circle with diameter

{AB}

. The bisectors of

{\angle BAC}

and

{\angle ABC}

intersect

{\Gamma}

(also) at

{D}

and

{E}

, respectively. The incircle of

{ABC}

meets

{BC}

and

{AC}

at

{F}

and

{G}

, respectively. Prove that

{D, E, F}

and

{G}

are collinear.

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