The Kariya problem and related constructions - Paul Yiu Given a point Q other than the incenter I of a reference triangle, we give a simple conic construction of a homothety mapping I into Q so that the image of the intouch triangle is perspective with the reference triangle. This is a generalization of the Kariya theorem in the case Q=I that the homothety can be arbitrary. The ratio of the homothety (the Kariya factor) is a unique nonzero finite number except when Q lies on the Feuerbach hyperbola or the line joining the incenter to the orthocenter of the reference triangle. For each nonzero real number t, we show that the locus of Q with Kariya factor t is a rectangular hyperbola. We give two simple constructions of this hyperbola. ✪ ✪ ✪ ✪ ✪ Link tải tài liệu: LINK TẢI TÀI LIỆU Theo LTTK Education
