5P-s-P4  5P-Keizer-Schmidt Center

This point was found by Bernard Keizer and Eckart Schmidt searching for the background of 5 mysterious points, being the 5 common points of the 3 QL-versions of the cubic QA-Cu7.
See Ref-34, QFG#3256.
Consider a 5P with circumconic Co
… and a line L connecting two 5P-vertices
… and the circumcircle Ci of the remaining vertices.
… Let X be the 4th intersection of Ci and Co,
… let L' be a parallel to L through X,
… intersecting Co further in Y,
… which will be a new 5P-point on the circumconic,
… independent of the choice of vertices.

Coordinates:
When using barycentric coordinates: P1=(1:0:0), P2=(0:1:0), P3=(0:0:1), P4=(p:q:r), P5=(P:Q:R),
then 5P-s-P4 has coordinates:
(p P/Ta : q Q/Tb : r R/Tc),
where:
Ta = +c2 p P2 q-c2 p2 P Q+b2 p P2 r+a2 P2 q r+b2 p P Q r-c2 p P Q r+a2 P q Q r
-b2 p2 P R-b2 p P q R+c2 p P q R-a2 p2 Q R-a2 p q Q R+a2 P q r R-a2 p Q r R,
Tb = -c2 P q2 Q+c2 p q Q2+b2 p P Q r+a2 P q Q r-c2 P q Q r+b2 p Q2 r+a2 q Q2 r
-b2 p P q R-b2 P q2 R-a2 p q Q R+c2 p q Q R-a2 q2 Q R-b2 P q r R+b2 p Q r R,
Tc = -c2 p P Q r-c2 P q Q r-c2 P Q r2+c2 p P q R+c2 p q Q R+a2 P q r R-b2 P q r R
-a2 p Q r R+b2 p Q r R-b2 P r2 R-a2 Q r2 R+c2 p q R2+b2 p r R2+a2 q r R2