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 also are 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.

The six-point circle is actually the circle with the midpoints of the two QG-diagonals as diameter.

Independently, Dario Pellegrinetti proved the concyclic property of the forementioned six points [68] and Eckart Schmidt showed that the Thales circle of the midpoints of the QG-diagonals passes through the Van Aubel points (personal note).

CT-Equation QG-Ci5 in 1^st QA-Quadrigon:

(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.

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