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Tangent circles !!!

Source: Romania TST 2015 Day 3 Problem 1

June 4, 2015
circlesInversionTangentsRomanian TSTgeometry

Problem Statement

Two circles γ\gamma and γ′\gamma' cross one another at points AA and BB . The tangent to γ′\gamma' at AA meets γ\gamma again at CC , the tangent to γ\gamma at AA meets γ′\gamma' again at C′C' , and the line CC′CC' separates the points AA and BB . Let Γ\Gamma be the circle externally tangent to γ\gamma , externally tangent to γ′\gamma' , tangent to the line CC′CC', and lying on the same side of CC′CC' as BB . Show that the circles γ\gamma and γ′\gamma' intercept equal segments on one of the tangents to Γ\Gamma through AA .
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