Perspective Isoconjugate Triangle Pairs, Hofstadter Pairs, and Crosssums on the Nine-Point Circle

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 ∗, P1, P2 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 ⊕ X ′, where ⊕ 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.