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equal segments related to 3 circumcircles

Source: Rioplatense Olympiad 2018 level 3 p2

December 11, 2018

geometrycircumcircleequal segmentssymmetry

Problem Statement

Let

P

be a point outside a circumference

\Gamma

, and let

PA

be one of the tangents from

P

to

\Gamma

. Line

l

passes through

P

and intersects

\Gamma

at

B

and

C

, with

\omega_2

between

P

and

C

. Let

D

be the symmetric of

B

with respect to

P

. Let

\omega_1

and

\omega_2

be the circles circumscribed to the triangles

DAC

and

PAB

respectively.

\omega_1

and

\omega _2

intersect at

E \neq A

. Line

EB

cuts back to

\omega _1

in

F

. Prove that

CF = AB

.

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