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two circles intersecting, prove that three lines are concurrent

Source: IGO 2021 Advanced P2

December 30, 2021

geometryIGOconcurrent

Problem Statement

Two circles

\Gamma_1

and

\Gamma_2

meet at two distinct points

A

and

B

. A line passing through

A

meets

\Gamma_1

and

\Gamma_2

again at

C

and

D

respectively, such that

A

lies between

C

and

D

. The tangent at

A

to

\Gamma_2

meets

\Gamma_1

again at

E

. Let

F

be a point on

\Gamma_2

such that

F

and

A

lie on different sides of

BD

, and

2\angle AFC=\angle ABC

. Prove that the tangent at

F

to

\Gamma_2

, and lines

BD

and

CE

are concurrent.

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