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midpoint M lies on line X_1X_2,also lies on Y_1Y_2 , interecting circles

Source: Ukrainian Geometry Olympiad 2020, X p3

April 27, 2020

geometrycirclesTangentsmidpoint

Problem Statement

The circles

\omega_1

and

\omega_2

intersect at points

A

and

B

, point

M

is the midpoint of

AB

. On line

AB

select points

S_1

and

S_2

. Let

S_1X_1

and

S_1Y_1

be tangents drawn from

S_1

to circle

\omega_1

, similarly

S_2X_2

and

S_2Y_2

are tangents drawn from

S_2

to circles

\omega_2

. Prove that if the point

M

lies on the line

X_1X_2

, then it also lies on the line

Y_1Y_2