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Incircle geometry

Source: Kvant Magazine No. 10 2021 M2672

March 9, 2023

geometryKvant

Problem Statement

Let the inscribed circle

\omega

of the triangle

ABC

have a center

I{}

and touch the sides

BC, CA

and

AB

at points

D, E

and

F{}

respectively. Let

M{}

and

N{}

be points on the straight line

EF

such that

BM \parallel AC

and

CN \parallel AB

. Let

P{}

and

Q{}

be points on the segments

DM{}

and

DN{}

, respectively, such that

BP \parallel CQ

. Prove that the intersection point of the lines

PF

and

QE

lies on

\omega

.
Proposed by Don Luu (Vietnam)

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