QG-Ci5: QG-Six-point Circle

The QG-Six-point circle is a circle through 6 main points especially related to the Van Aubel configuration. These six points are:
• The midpoints of the two QG-diagonals Mac and Mbd
• The Inner and Outer Van Aubel Points QG-2P6a and QG-2P6b, which are also the centers of the only possible circumscribed squares about a Quadrigon.
• The common midpoints of the Inner and Outer Van Aubel Segments Mseg1 and Mseg2.
Independently, Dario Pellegrinetti proved the concyclic property of the forementioned six points (Ref-68) and Eckart Schmidt showed that the Thales circle of the midpoints of the QG-diagonals passes through the Van Aubel points (personal note).

(a2 p - b2 p - c2 p + 2 a2 q - 2 b2 q - 2 c2 q + a2 r - 3 b2 r - c2 r) x2
+ (-a2 p + b2 p - 3 c2 p - 3 a2 r + b2 r - c2 r) y2
+ (-a2 p - 3 b2 p + c2 p - 2 a2 q - 2 b2 q + 2 c2 q - a2 r - b2 r + c2 r) z2
+ 2 (2 c2 p + a2 q - b2 q + 3 c2 q - a2 r - b2 r + 3 c2 r) x y
+ 2 (3 a2 p - b2 p - c2 p + 3 a2 q - b2 q + c2 q + 2 a2 r) y z
+ 4 b2 (p + q + r) x z

Properties:

• The center of QG-Ci5 is the Quadrangle Centroid QA-P1.
• The circle is the locus of centers of all circumscribed rectangles about a Quadrigon.
QG-Ci5 is the circle with diameter Mac.Mbd as well as the circle with diameter Mseg1.Mseg2.

Vernieuwen