Paul Yiu, The Kariya problem and related constructions,
Forum Geometricorum, 15 (2015) 191--201.
Abstract. 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.
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