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MP = NQ wanted, incircles related

Source: IMO 2019 SL G2

September 22, 2020

geometryIMO ShortlistIMO Shortlist 2019similar trianglestrigonometryCanadaTST

Problem Statement

Let

AB

be an acute-angled triangle and let

D, E

, and

F

be the feet of altitudes from

A, B

, and

C

to sides

BC, CA

, and

AB

, respectively. Denote by

\omega_B

and

\omega_C

the incircles of triangles

BDF

and

CDE

, and let these circles be tangent to segments

DF

and

DE

at

M

and

N

, respectively. Let line

MN

meet circles

\omega_B

and

\omega_C

again at

P \ne M

and

Q \ne N

, respectively. Prove that

MP = NQ

.
(Vietnam)

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