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3 collinear leads to 4 collinear , incircle related

Source: Ukraine TST 2011 p6

May 5, 2020

geometrycollinearincircle

Problem Statement

The circle

\omega

inscribed in triangle

M

touches its sides

AB, BC, CA

at points

K, L, M

respectively. In the arc

KL

of the circle

\omega

that does not contain the point

M

, we select point

S

. Denote by

P, Q, R, T

the intersection points of straight

AS

and

KM, ML

and

SC, LP

and

KQ, AQ

and

PC

respectively. It turned out that the points

R, S

and

M

are collinear. Prove that the point

T

also lies on the line

SM

.

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