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 [34], QFG#3256.

Construction

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

Application in Quadrilateral Geometry

A new Quadrangle Transformation QA-Tfx can be made, which considers the reference points of the Quadrangle and a random point P as the 5P (Pentangle) delivering the new 5P-point 5P-s-P4.

Properties of this Quadrangle Transformation QA-Tfx are:

• QA-Tfx (QA-P4) = QA-P1.QA-P42 ^ QA-P3.QA-P4 ^ QA-P6.QA-P34.
• QA-Tfx (QA-P41) lies on QA-P3.QA-P41.
• For a point P on QA-Co3 the image QA-Tfx(P) lies diametral on QA-Co3. See [34], QFG-message #3256 and #3424.

Properties

• In a Triangle 5P-s-P4 of Pentangle ABCBr1Br2 (where Br1, Br2 are the 2 Brocard Points of Triangle ABC) is ETC Center X(99). See [12].
• In a Reference Quadrilateral we have 3 Component Quadrigons (see QL-3QG1). For each Component Quadrigon we can construct the circular cubic QA-Cu1. These three cubics have 5 common points, which form a Pentangle extensively discussed in the Quadri-Figures Group, see [34], keywords “3 QL-versions of QA-Cu7” and “5P-Geometry”. The point 5P-s-P4 of this Pentangle coincides a point T, being the intersection point of the 3 QL-versions of QA-Px.QG-P1 of the Reference Quadrilateral, where QA-Px = QA-P41.QG-P19 ^ QA-P4.QG-P18.

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