Given a triangle Δ A B C and the excentral triangle Δ J_A J_B J_C, define the A'-vertex of the hexyl triangle as the point in which the perpendicular to A B through the excenter J_B meets the perpendicular to A C through the excenter J_C, and similarly define B' and C'. Then Δ A' B' C' is known as the hexyl triangle of Δ A B C, and A' J_B C' J_A B' J_C forms a hexagon with parallel sides (Kimberling 1998 pp. 79 and 172). The hexyl triangle has trilinear vertex matrix [x + y + z + 1 | x + y - z - 1 | x - y + z - 1 x + y - z - 1 | x + y + z + 1 | -x + y + z - 1 x - y + z - 1 | -x + y + z - 1 | x + y + z + 1], where x = cos A, y = cos B, and z = cos C.

excentral-hexyl ellipse | excentral triangle | hexyl circle