QA-Tf13: QA-AntiPedal Point

Definition: The AntiPedal triangle of P wrt a reference triangle ABC is the triangle of which ABC is the Pedal Triangle of P (see QA-Tf12). The vertices of the AntiPedal Triangle are called the AntiPedal Points.

The QA-AntiPedal Point is defined as follows.

The 3 AntiPedal Points of P wrt a Triangle define a AntiPedal Circle Ci.

In a Quadrangle the 4 AntiPedal Circles Ci (denoted as Ci1, Ci2, Ci3, Ci4) of the Component Triangles intersect at a common point being the QA-AntiPedal Point.

This transformation is mentioned first at [34], QFG#1223.

See also [34], QFG-message #2759 for a generalization to n-Points.

1^st CT-coordinate of QA-Tf13[(x:y:z)]:

Formula of the 15^th degree. Too long and too complicated to be mentioned here.

Properties

• QA-P4 is transformed into QA-P9. See [34], QFG#1223.
• When P is any point on a Circumscribed Circle of a QA-Component Triangle, then QA-Tf13(P) will be the point diametrically to P on that circle.
• QA-Tf13 can be generalized as the transformation nP-e-Tf2 in an n-Point (system of n reference points). It matches with nP-e-Tf2 when n=4. See [34], QFG#2759.

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