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Iran geometry

Source: Iranian TST 2018, first exam, day1, problem 3

April 7, 2018

geometry

Problem Statement

In triangle

ABC

let

M

be the midpoint of

BC

. Let

\omega

be a circle inside of

P,Q

and is tangent to

AB,AC

at

E,F

, respectively. The tangents from

M

to

\omega

meet

\omega

at

P,Q

such that

P

and

B

lie on the same side of

AM

. Let

X \equiv PM \cap BF

and

Y \equiv QM \cap CE

. If

2PM=BC

prove that

XY

is tangent to

\omega

.
Proposed by Iman Maghsoudi

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