5P-s-P6 5P-Miquel Point

Consider a 5-Point (Pentangle) with vertices Pi (i=1,2,3,4,5).
Let Ri = QA-Tf16(Pi) wrt QA=5-Point minus Pi.
Let Rj = QA-Tf16(Pj) wrt QA=5-Point minus Pj.
Now 5P-s-P6 = QL-P1 of the QL with lines Pi.Pj, Pi.Rj, Pj.Ri, Ri.Rj.
This point is fixed and independent of the chosen 5-Point-vertices Pi, Pj and therefore it is a regular 5P-Center.
This point was found by Eckart Schmidt. See Ref-34, QFG #3270, #3583, #3588.

5P s P6 5P Miquel Point 02


• In a Triangle 5P-s-P6 of Pentangle ABCBr1Br2 (where Br1, Br2 are the 2 Brocard Points of Triangle ABC) is ETC Center X(2715). 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 finite common points, which form a Pentangle extensively discussed in the Quadri-Figures Group, see Ref-34, keywords “3 QL-versions of QA-Cu7” and “5P-Geometry”.
The point 5P-s-P6 of this Pentangle coincides with the Miquel point QL-P1 of the Reference Quadrilateral.