Peter J. C. Moses and Clark Kimberling, Perspective isoconjugate triangle pairs, Hofstadter pairs, and crosssums on the nine-point circle,
Forum Geometricorum, 11 (2011) 83--93.

Abstract.   The r-Hofstadter triangle and the (1-r)-Hofstadter triangle are proved perspective, and homogeneous trilinear coordinates are found for the perspector. More generally, given a triangle DEF inscribed in a reference triangle ABC, triangles A'B'C' and A''B''C'' derived in a certain manner from DEF are perspective to each other and to ABC. Trilinears for the three perspectors, denoted by P*, P_1, P_2 are found (Theorem 1) and used to prove that these three points are collinear. Special cases include (Theorems 4 and 5) this: if X and X' are an antipodal pair on the circumcircle, then the perspector P* = X \oplus X', where \oplus denotes crosssum, is on the nine-point circle. Taking X to be successively the vertices of a triangle DEF inscribed in the circumcircle thus yields a triangle D'E'F' inscribed in the nine-point circle. For example, if DEF is the circumtangential triangle, then D'E'F' is an equilateral triangle.

[ps file] [pdf]