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Iran Geometry

Source: Iran MO 3rd round 2017 finals - Geometry P1

September 3, 2017

geometrycircumcircle

Problem Statement

Let

ABC

be a right-angled triangle

\left(\angle A=90^{\circ}\right)

and

M

be the midpoint of

BC

.

C,M

is a circle which passes through

B,M

and touchs

AC

at

X

.

\omega_2

is a circle which passes through

C,M

and touchs

AB

at

Y

(

X,Y

and

A

are in the same side of

BC

). Prove that

XY

passes through the midpoint of arc

BC

(does not contain

A

) of the circumcircle of

ABC

.

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