5P-s-P1: 5P-Circumscribed Conic Center

It is well known that in a system of 5 random Points a unique circumscribed conic can be constructed.
This conic is 5P-s-Co1 and its center is 5P-s-P1.

Construction (See Ref-19):
1. Let the conic be defined by points A, B, C, D, E.
2. Let the tangents at A, B meet at T, and those at B, C meet at TO.
3. Let M, MO be the midpoints of AB and BC, then the center O is MT.MOTO.
Construction of Conic Tangents:
4. Let d = AB, e = BC, a = CD, b = DE, c = EA, then bd.ce cuts a in a point lying on the tangent at A.

5P s P1 Center Circumscribed Conic 01

When using barycentric coordinates: P1=(1:0:0), P2=(0:1:0), P3=(0:0:1), P4=(p:q:r), P5=(x:y:z),
then 5P-s-P1 has coordinates:
          (-p P (Q r - q R) (p P Q r + P q Q r - p P q R - p q Q R - P q r R + p Q r R) :
           -q Q (P r - p R) (p P Q r + P q Q r - p P q R - p q Q R + P q r R - p Q r R) :
           -(P q - p Q) r R (-p P Q r + P q Q r + p P q R - p q Q R + P q r R - p Q r R))

Correspondence with ETC/EQF:
Application in Triangle Geometry:
  • 5P-s-P1 of Pentangle ABCBr1Br2 (where Br1, Br2 are the 2 Brocard Points of Triangle ABC) is ETC Center X(8290). See Ref-12.

5P-s-Tf3(5P-s-P1) = 5P-s-P1.
5P-s-P1 is the common point of the radical axes of the 5 versions of QA-Ci1 (Circumcircle of the Diagonal Triangle) in the 5-Point.
5P-s-P1 is the common point of the 5 versions of QA-Co1 (Nine Point Conic) in the 5-Point.

5P s P1 QA Co1 NPC Common Point 01