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Collinear points and incircle

Source: Kvant Magazine No. 2 2022 M2688

March 8, 2023

Kvantgeometry

Problem Statement

Let

T_a, T_b

and

T_c

be the tangent points of the incircle

\Omega

of the triangle

ABC

with the sides

BC, CA

and

AB{}

respectively. Let

X, Y

and

Z{}

be points on the circle

\Omega

such that

A{}

lies on the ray

YX

,

B{}

lies on the ray

ZY

and

C{}

lies on the ray

XZ

. Let

P{}

be the intersection point of the segments

ZX

and

T_bT_c

, and similarly

Q=XY \cap T_cT_a

and

R=YZ\cap T_aT_b

. Prove that the points

A, B

and

C{}

lie on the lines

RP, PQ

and

QR{}

, respectively.
Proposed by L. Shatunov (11th grade student)

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