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Suppose O,K,L are collinear

Source: All-Russian MO 2000

December 30, 2012

trigonometrygeometrygeometric transformationreflectionangle bisector

Problem Statement

A quadrilateral

ABCD

is circumscribed about a circle

\omega

. The lines

AB

and

CD

meet at

O

. A circle

\omega_1

is tangent to side

BC

at

K

and to the extensions of sides

AB

and

CD

, and a circle

\omega_2

is tangent to side

AD

at

L

and to the extensions of sides

AB

and

CD

. Suppose that points

O

,

K

,

L

lie on a line. Prove that the midpoints of

BC

and

AD

and the center of

\omega

also lie on a line.

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