Some Properties of the Exeter Transformation

The Exeter point of a given triangle <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>B</mi><mi>C</mi></mrow></semantics></math></inline-formula> 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>B</mi><mi>C</mi></mrow></semantics></math></inline-formula>, 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.