# euclidean geometry – Six concyclic points

Proposition. Let $$triangle ABC$$ be an arbitrary triangle with excenters $$J_A$$,$$J_B$$ and $$J_C$$ . Let $$G$$ be the orthogonal projection of the $$J_B$$ on the extension of the side $$BC$$ , $$H$$ orthogonal projection of the $$J_B$$ on the extension of the side $$AB$$ , $$I$$ orthogonal projection of the $$J_C$$ on the extension of the side $$AC$$ , $$J$$ orthogonal projection of the $$J_C$$ on the extension of the side $$BC$$ , $$K$$ orthogonal projection of the $$J_A$$ on the extension of the side $$AB$$ and $$L$$ orthogonal projection of the $$J_A$$ on the extension of the side $$AC$$ . Now let $$M$$ be the point of intersection of the line segments $$GH$$ and $$J_AJ_B$$ ,$$N$$ point of intersection of the line segments $$GH$$ and $$J_BJ_C$$ , $$O$$ point of intersection of the line segments $$IJ$$ and $$J_BJ_C$$ , $$P$$ point of intersection of the line segments $$IJ$$ and $$J_AJ_C$$ , $$Q$$ point of intersection of the line segments $$LK$$ and $$J_AJ_C$$ and $$R$$ point of intersection of the line segments $$LK$$ and $$J_AJ_B$$ . I claim that the points $$M$$,$$N$$,$$O$$,$$P$$,$$Q$$,$$R$$ lie on a common circle.

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