TR-Tf4 Triangular Perspective Center
The Triangular Perspective Center of two points P and R with respect to a Reference Triangle ABC is the Perspective Center M for which P is the Perspective Conjugate (TR-Tf3) of R, and conversely, R is the Perspective Conjugate of P.
Clarification
The Triangular Perspective Center of two points P and R with respect to a Reference Triangle ABC is the Perspective Center M for which, on each sideline, the traces of P and R as well as with the corresponding pair of vertices induce a perspectivity whose center is the trace of M and whose vanishing point is the intersection of that sideline with the M-Trilinear Polar.
Construction
The following figure illustrates the construction.
The next figure provides a construction for locating the point ππ (and, by analogy, ππ and ππ).
Note, that in determining the pair (Perspective Midpoint ππ, Vanishing Point ππ), the harmonic conjugate is used as the relevant perspectival transformation, since (according to PR-Tf1) ππ is the harmonic conjugate of ππ with respect to (π΅,πΆ).
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