Abstract
The Exeter point of a given triangle ABC is the center of perspective of the tangential triangle and the circummedial triangle of the given triangle. The process of the Exeter point from the centroid serves as a base for defining the Exeter transformation with respect to the triangle ABC, which maps all points of the plane. We show that a point, its image, the symmedian, and three exsymmedian points of the triangle are on the same conic. The Exeter transformation of a general line is a fourth-order curve passing through the exsymmedian points. We show that each image point can be the Exeter transformation of four different points. We aim to determine the invariant lines and points and some other properties of the transformation.
Highlights
With the help of the barycentric coordinates, we showed that the extension of the wellknown process of the Exeter point from the centroid of a given triangle ABC provides a socalled Exeter transformation for the whole plane
Each point P, its image Pe, the symmedian, and three exsymmedian points of the triangle are on the same conic Q( P)
The Exeter transformation is nonlinear, as the image of a general line is a fourth-order curve passing through the exsymmedian points of the triangle
Summary
The Exeter point of a given triangle is defined from the centroid of the triangle by a drawing process. As a generalization of the definition of the Exeter point to the whole plane of the triangle, we define a so-called Exeter transformation with respect to a given triangle ABC. Minevich and Morton [2] defined a similar, so-called “TCC-perspector”, transformation with respect to 4 ABC, and they gave a nice connection between the isogonal transformation and the “TCC-perspector”. For verifying our statements we use an analytical way with barycentric coordinates. The base triples of this barycentric coordinate system we use the vertices of a given triangle. There are many interesting articles dealing with the use of barycentric coordinates, and among them the works in [7,8,9] may be useful
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