Perspective isoconjugate triangle pairs, Hofstadter pairs, andcrosssums on the nine-point circle

摘要

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.