TR-Tf2 Trilinear Pole

The Trilinear Pole of a line L with respect to a reference triangle ABC is the perspector of the cevian triangle of M and ABC. It is a transformation that maps a line, in the context of a triangle, to a point.

From a perspectival viewpoint, the trilinear pole is the perspective center generated by the perspectivities arising on each sideline of the triangle. For each sideline, the corresponding pair of vertices, together with the intersection point Vx (x=a,b,c) of that sideline with L — functioning as a vanishing point — determine a perspective scale with an associated perspective midpoint Mx (x=a,b,c). The three perspective midpoints Mx (x=a,b,c) obtained according to (PR-Tf2) are the traces of a point M, which is the trilinear pole of L.

Construction

The following figure illustrates the construction.

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