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Lines thru the midpoint of the common chord of two circles

Source: XVIII Tuymaada Mathematical Olympiad (2011), Senior Level

July 29, 2011

geometrygeometric transformationreflectiongeometry unsolved

Problem Statement

Circles

\omega_1

and

\omega_2

intersect at points

A

and

B

, and

M

is the midpoint of

AB

. Points

S_1

and

S_2

lie on the line

AB

(but not between

A

and

B

). The tangents drawn from

S_1

to

\omega_1

touch it at

X_1

and

Y_1

, and the tangents drawn from

S_2

to

\omega_2

touch it at

X_2

and

Y_2

. Prove that if the line

X_1X_2

passes through

M

, then line

Y_1Y_2

also passes through

M

.

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