Wallace-Simson Line in Four Cayley-Klein Planes

The Wallace-Simson line is the line containing the feet of the lines perpendicular from an arbitrary point on the circumcircle of a triangle to the sides of the triangle. The line was attributed to R. Simson by J. V. Poncelet, but today it is usually known as the Wallace-Simson line since the term was introduced by W. Wallace in 1797 and it does not actually appear in any work of Simson. Various interesting properties are known such as that the set of all of the Wallace-Simson lines for a given triangle form an envelope of a deltoid which is known as the Steiners deltoid. In this work we will treat the Wallace-Simson line and Steiners deltoid, except in the Euclidean plane, as well as in three other Cayley-Klein planes: the quasi-elliptic, the pseudo-Euclidean and the quasi-hyperbolic plane.