QG-P12: Inscribed Harmonic Conic Center

The Inscribed Harmonic Conic Center is the Center of the Inscribed Harmonic Conic QG-Co1. This conic touches the sidelines of the Quadrigon in their perspective midpoints.
See picture below. Coordinates:
• (p(2p+q+r) : q(p-r) : r(p-q))
• (p(r-q) : q (r-p) : r(p+q+2r))
• (p(q-r) : q(p+2q+r) : r(q-p))
• (   m + n   : -2l + n : -2l + m)
• (-2m + n  :     l + n : -2m + l)
• ( -2n + m : -2n + l :     l + m)
CT-Area of QG-P12–Triangle in the QA-environment:
• p q r (p + q)(p + r)(q + r) Δ / ((p2 + p q + p r - q r) (p q + q2 - p r + q r) (p q - p r - q r - r2))
CT-Area of QG-P12–Triangle in the QL-environment:
• 0     (points are collinear on Newton Line)

• (-p2 : q2 :  r2)
• ( p2 : -q2 : r2)
• ( p2 : q2 : -r2)
• (-2 m2 n2 :     l2 n2 :       l2 m2)
• (     m2 n2 : -2 l2 n2 :     l2 m2)
• (     m2 n2 :      l2 n2 : -2 l2 m2)
DT-Area of QG-P12–Triangle in the QA-environment:
• (-2 S p2 q2 r2) / ((-p2+q2+r2) (p2+q2-r2) (p2-q2+r2))
DT-Area of QG-P12–Triangle in the QL-environment:
• 0     (points are collinear on Newton Line)

Properties:
• QG-P12 is the fourth harmonic point of QG-P2 (Midpoint 3rd Diagonal) on the Newton Line (QL-L1) wrt the midpoints of the diagonals (note Eckart Schmidt).
• QG-P12 is collinear with QG-P1, QG-P13, QA-P16, QL-P13 on QG-L2.
• QG-P12 is collinear with QA-P42 and QL-P26.
• The triangle formed by the 3 QA-versions of QG-P12 is perspective with the QA-Diagonal Triangle. Their Perspector is QA-P16 (QA-Harmonic Center).
• The triangle formed by the 3 QL-versions of QG-P12 is a flat Triangle (the 3 vertices are collinear on the Newton Line QL-L1) and is perspective with the QL-Diagonal Triangle. Their Perspector is QL-P13 (QL-Harmonic Center).
• QG-P12 is the intersection point of the Trilinear Polars of QG-P13 as well as QL-P13 wrt the QA-Diagonal Triangle QA-Tr1 and the QL-Diagonal Triangle QL-Tr1 (note Eckart Schmidt).
• The Involutary Conjugate   (QA-Tf2) of QG-P12 is the Reflection of QG-P1 in QG-P2.
• The involutary Conjugate   (QA-Tf2) of QG-P12 lies on the line QG-P14.QG-P15.QA-P5.
• The Involutary Conjugates (QA-Tf2) of QG-P12 and QG-P13 both lie on the line QG-P1.QG-P2.QA-P10.
• The 3 QA-versions of QG-P12 lie on the QA-circumconic through QA-P1 (Eckart Schmidt, October 9, 2013). See Ref-34, QFG #286,287.

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