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Mehmet Efe Akengin, Zeyd Yusuf Koroglu, and Yigit Yargic, Three natural homoteties of the nine-point circle,
Forum Geometricorum, 13 (2013) 209--218.
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Abstract. Given a triangle with the reflections of its vertices in the opposite sides, we prove that the pedal circles of these reflections are the images of nine-point circle under specific homoteties, and that their centers form the anticevian triangle of the nine-point center. We also construct two concentric circles associated with the pedals of these reflections on the sidelines, and study the triangle bounded by the radical axes of these pedal circles with the nine-point circle.

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