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parallel wanted, starting with equal tangent circles

Source: 2018 Indonesia MO Province P2 q3 OSP

December 10, 2020

geometryparallelequal circles

Problem Statement

Let

\Gamma_1

and

\Gamma_2

be two different circles with the radius of same length and centers at points

O_1

and

O_2

, respectively. Circles

\Gamma_1

and

\Gamma_2

are tangent at point

\Gamma_1

. The line

\ell

passing through

O_1

is tangent to

A

at point

A

. The line

M

intersects

AX

at point

X

with

X

between

A

and

O_1

. Let

M

be the midpoint of

AX

and

Y

the intersection of

PM

and

\Gamma_2

with

Y\ne P

. Prove that

XY

is parallel to

O_1O_2

.

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