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Reflection about the sides

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December 9, 2010
geometrygeometric transformationreflectioncircumcircleEuleranalytic geometrygeometry unsolved

Problem Statement

Suppose △ABC\triangle ABC has circumcircle Γ\Gamma, circumcentre OO and orthocentre HH. Parallel lines α,β,γ\alpha, \beta, \gamma are drawn through the vertices A,B,CA, B, C, respectively. Let α′,β′,γ′\alpha ', \beta ', \gamma ' be the reflections of α,β,γ\alpha, \beta, \gamma in the sides BC,CA,ABBC, CA, AB, respectively.
(a)(a) Show that α′,β′,γ′\alpha ', \beta ', \gamma ' are concurrent if and only if α,β,γ\alpha, \beta, \gamma are parallel to the Euler line OHOH.
(b)(b) Suppose that α′,β′,γ′\alpha ', \beta ' , \gamma ' are concurrent at the point PP . Show that Γ\Gamma bisects OPOP .
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