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Concurrent lines in circles tangent to circumcircle diagram

Source: XII Rioplatense Mathematical Olympiad (2003), Level 3

August 8, 2011

geometrycircumcirclegeometric transformationvideosprojective geometrygeometry unsolvedmixtilinear incircle

Problem Statement

Triangle

ABC

is inscribed in the circle

\Gamma

. Let

\Gamma_a

denote the circle internally tangent to

\Gamma

and also tangent to sides

AB

and

AC

. Let

A'

denote the point of tangency of

\Gamma

and

\Gamma_a

. Define

B'

and

C'

similarly. Prove that

AA'

BB'

and

CC'

are concurrent.