Let ABC be a triangle.
1. Let (N1),(N2),(N3) be the reflections of the NPC (N) in the sidelines BC,CA,AB, resp. and A'B'C' the triangle bounded by the radical axes of ((O),(N1)), ((O),(N2)), ((O),(N3)), resp.
The triangles ABC, A'B'C' are perspective.
2. Let HaHbHc be the orthic triangle, (N1),(N2),(N3) the reflections of the NPC (N) in the altitudes HHa,HHb,HHc, resp. and and A'B'C' the triangle bounded by the radical axes of ((O),(N1)), ((O),(N2)), ((O),(N3)), resp.
The triangles HaHbHc, A'B'C' are perspective.
The triangles ABC, A'B'C' are perspective.
2. Let HaHbHc be the orthic triangle, (N1),(N2),(N3) the reflections of the NPC (N) in the altitudes HHa,HHb,HHc, resp. and and A'B'C' the triangle bounded by the radical axes of ((O),(N1)), ((O),(N2)), ((O),(N3)), resp.
The triangles HaHbHc, A'B'C' are perspective.
3. Let OaObOc be the medial triangle [pedal tr. of O], (N1),(N2),(N3) the reflections of the NPC (N) in the perp. bisectors OOa,OOb,OOc, resp. and and A'B'C' the triangle bounded by the radical axes of ((O),(N1)), ((O),(N2)), ((O),(N3)), resp.
The triangles ABC, A'B'C' are perspective.
Antreas P. Hatzipolakis, 11 Febr. 2013
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