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.

Coordinates:
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: