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Prove tangency

Source: Mathematics Regional Olympiad of Mexico Southeast 2018 P5

October 24, 2021

geometrycircumcircle

Problem Statement

Let

ABC

an isosceles triangle with

CA=CB

and

\Gamma

it´s circumcircle. The perpendicular to

A

through

B

intersect

\Gamma

in points

B

and

E

. The parallel to

BC

through

A

intersect

EC, \Omega

in points

A

and

D

. Let

F

the intersection of

ED

and

BC, I

the intersection of

BD

and

EC, \Omega

the cricumcircle of the triangle

ADI

and

\Phi

the circumcircle of

BEF

.If

O

and

P

are the centers of

\Gamma

and

\Phi

, respectively, prove that

OP

is tangent to

\Omega

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