There is an alternating sequence of circles & circle centers in n-Points, when n=odd and common circle points in n-Points, when n=even.
nP-e-Tf2 is a generic designation, denoting the specific transformations 4P-e-Tf2, 6P-e-Tf2, 8P-e-Tf2, etc.. They are point to point transformations in an n-Point for n=even.
nP-o-Tf3 is a generic designation, denoting the specific transformations 3P-o-Tf3, 5P-o-Tf3, 7P-o-Tf3, etc.. They are point to circle transformations in an n-Point for n=odd.
nP-o-Tf4 is a generic designation, denoting the specific transformations 3P-o-Tf4, 5P-o-Tf4, 7P-o-Tf4, etc.. They are point to point transformations in an n-Point for n=odd.
nP-o-Tf3 is a generic designation, denoting the specific transformations 3P-o-Tf3, 5P-o-Tf3, 7P-o-Tf3, etc.. They are point to circle transformations in an n-Point for n=odd.
nP-o-Tf4 is a generic designation, denoting the specific transformations 3P-o-Tf4, 5P-o-Tf4, 7P-o-Tf4, etc.. They are point to point transformations in an n-Point for n=odd.
Definition: The AntiPedal triangle of P wrt a reference triangle ABC is the triangle of which ABC is the Pedal Triangle of P. The vertices of the AntiPedal Triangle are called the AntiPedal Points.
The development of nP-o-Tf3, nP-o-Tf4, nP-e-Tf2 for increasing n is best understood by starting with n=3.
• 3-Point: The 3 AntiPedal Points of P wrt a 3-Point define a Circle 3P-o-Tf3 with center 3P-o-Tf4.
• 4-Point: The 4 circles 3P-o-Tf3 of the component triangles intersect at a common point 4P-e-Tf2.
• 5-Point: The 5 common points 4P-e-Tf2 of the component 4-Points lie on a circle 5P-o-Tf3 with center 5P-o-Tf4.
• 6-Point: The 6 circles 5P-o-Tf3 of the component 5-Points intersect at a common point 6P-e-Tf2.
• 7-Point: The 7 common points 6P-e-Tf2 of the component 6-Points lie on a circle 7P-o-Tf3 with center 7P-o-Tf4.
• etc.
For more background on the transformation see nP-e-Tf1.
Correspondence with ETC/EQF:
In a 3-Point-configuration:
3P-o-Tf3 = P-AntiPedal Circle in a Triangle
3P-o-Tf4 = P-AntiPedal Circle Center in a Triangle
In a 4-Point-configuration:
4P-e-Tf1 = QA-Tf13
3P-o-Tf3 = P-AntiPedal Circle in a Triangle
3P-o-Tf4 = P-AntiPedal Circle Center in a Triangle
In a 4-Point-configuration:
4P-e-Tf1 = QA-Tf13
Properties:
• When n = 5 and P lies on the 5P-Circumscribed Conic 5P-s-Co1, then P also will lie on the 5P-AntiPedal Circle 5P-o-Tf3(P).
• The 4 versions of 3P-o-Tf4(Pi)-Circlecenters (3P-o-Tf4(Pi) applied to triangles PjPkPl) form a Quadrangle QA1 which is similar to the Reference Quadrangle QA0. QA0 and QA1 share the same point QA-P4.