Let ABC be a triangle and let I and O denote its incentre and circumcentre respectively. Let ωA be the circle through B and C which is tangent to the incircle of the triangle ABC; the circles ωB and ωC are defined similarly. The circles ωB and ωC meet at a point A′ distinct from A; the points B′ and C′ are defined similarly. Prove that the lines AA′,BB′ and CC′ are concurrent at a point on the line IO.(Russia) Fedor Ivlev geometryincentercircumcirclegeometric transformationreflectionhomothetytrigonometry