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P, N, L collinear if KM//AC, incircle and a circle 2021 Kharkiv City MO 10.5

Source:

December 29, 2021

geometrycollinearincircle

Problem Statement

The inscribed circle

\Omega

of triangle

ABC

touches the sides

AB

and

AC

at points

K

and

L

, respectively. The line

BL

intersects the circle

\Omega

for the second time at the point

M

. The circle

\omega

passes through the point

M

and is tangent to the lines

AB

and

BC

at the points

P

and

Q

, respectively. Let

N

be the second intersection point of circles

\omega

and

\Omega

, which is different from

M

. Prove that if

KM \parallel AC

then the points

P, N

and

L

lie on one line.

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