Reflection Triangle

The reflection triangle is the triangle obtained by reflecting the vertices of a reference triangle about the opposite sides (Grinberg 2003). It is perspective to the reference triangle with the orthocenter as the perspector, and has trilinear vertex matrix

(1)

Its side lengths are

			(2)
			(3)
			(4)

Its area is given by

			(5)
			(6)

(P. Moses, pers. comm., Jan. 31, 2005), where is the circumcenter, is the orthocenter, is the circumradius, and is the area of the reference triangle.

Its triangle centroid has triangle center function

			(7)
			(8)

which is not a Kimberling center (P. Moses, pers. comm., Feb. 7, 2005), where , , , and are Conway triangle notation. The circumcircle of the reflection triangle is the reflection circle, and its circumcenter is Kimberling center , which is the -Ceva conjugate of . Its orthocenter has a complicated triangle center function that is not a Kimberling center.

The reflection triangle is perspective to the Cevian triangles with Cevian points lying on the orthopivotal cubic curve K060, corresponding to Kimberling centers for , 5, 13, 14, 30, 79, 80, 621, 622, 1117, and 1141. It is perspective to the anticevian triangles with anticevian points lying on the Napoleon-Feuerbach cubic, corresponding to Kimberling centers with , 3, 4, 5, 17, 18, 54, 61, 62, 195, 627, 628, 2120, and 2121. It is also perspective to the antipedal triangles with antipedal points corresponding to Kimberling centers with , 5, 20, 24, 54, 64, 68, 155, 254, and 2917 (P. Moses, pers. comm., Feb. 3, 2005).

The reflection triangle is degenerate iff

(9)

(Bottema 1987).

The reflection triangle is homothetic to the pedal triangle of the nine-point circle (Bottema 1987). In particular, if is the triangle centroid of , then the reflection triangle is the image of the pedal triangle of the nine-point center under the homothecy (Boutte 2001, cited in Grinberg 2003).

The circumcenter of the reflection triangle is Kimberling center , which is the -Ceva conjugate of .