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equal angles, interior point of triangle, concyclic

Source: Czech-Polish-Slovak Match 2017 day 1 P2

September 28, 2017

Concyclicequal anglesgeometrycircle

Problem Statement

Let

{\omega}

be the circumcircle of an acute-angled triangle

{ABC}

. Point

{D}

lies on the arc

{BC}

of

{\omega}

not containing point

{A}

. Point

{E}

lies in the interior of the triangle

{ABC}

, does not lie on the line

{AD}

, and satisfies

{\angle DBE =\angle ACB}

and

{\angle DCE = \angle ABC}

. Let

{F}

be a point on the line

{AD}

such that lines

{EF}

and

{BC}

are parallel, and let

{G}

be a point on

{\omega}

different from

{A}

such that

{AF = FG}

. Prove that points

{D,E, F,G}

lie on one circle.
(Slovakia)

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