MathDB

Problem 7

Source: 239-School Open Olympiad (Senior Level)

April 25, 2022

geometrytangent

Problem Statement

Points

A,B,C

are chosen inside the triangle

A_{1}B_{1}C_{1},

so that the quadrilaterals

B_{1}CBC_{1}, C_{1}ACA_{1}

and

A_{1}BAB_{1}

are inscribed in the circles

\Omega _{A}, \Omega _{B}

and

\Omega _{C},

respectively. The circle

Y_{A}

internally touches the circles

\Omega _{B}, \Omega _{C}

and externally touches the circle

\Omega _{A}.

The common interior tangent to the circles

Y_{A}

and

\Omega _{A}

intersects the line

BC

at point

A'.

Points

B'

and

C'

are analogously defined. Prove that points

A',B'

and

C'

are lying on the same line.

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