QG-P14 Center of the M3D Hyperbola

QG-P14 is the center of the M3D Hyperbola (see paragraph QG-Co3).
M3D stands for “Midpoint 3rd QA-Diagonal”.

Coordinates:
• (p (q + r) (-p q + p r + 3 q r + r2) :   q (p - r)2 (p + 2 q + r) : (p + q) r (p2 + 3 p q + p r - q r))
• (p (q + r) (p q + q2 - p r + 3 q r)   :   q (p + r) (p2 + p q + 3 p r -q r)   : (p - q)2 r (p + q + 2 r))
• (p (q - r)2 (2 p + q + r)  :  q (p + r) (-p q + 3 p r+- q r + r2)  : (p + q) r (3 p q + q2 - p r + q r))

• (-l n (-2 l m2 + l m n + 2 m2 n + l n2 - m n2)  :  m (l - n) (l2 m - l2 n - 2 l m n - l n2 + m n2)  :  l n (-l2 m + 2 l m2 + l2 n + l m n - 2 m2 n))
• ( l m (l m2 + l m n - m2 n - 2 l n2 + 2 m n2)    :  -l m (l2 m - l2 n + l m n + 2 l n2 - 2 m n2)  :  (l - m) n (l2 m + l m2 - l2 n + 2 l m n - m2 n))
• ( l (m - n) (l m2 - 2 l m n - m2 n + l n2 - m n2)  :  -m n (-2 l2 m + 2 l2 n + l m n - l n2 + m n2)  :  m n (2 l2 m - l m2 - 2 l2 n + l m n + m2 n))
CT-Area of QG-P14-Triangle in the QA-environment:
• 16 p q r (p2 + p q + p r - q r) (p q + q2 - p r + q r) (p q - p r -q r - r2) Δ  /  ((p + q)3 (p + r)3 (q + r)3)
CT-Area of QG-P14-Triangle in the QL-environment:       (equals 3 x area QL-Diagonal Triangle)
• 12 l2 m2 n2 Δ / ((-l m + l n + m n) (l m + l n - m n) (l m - l n + m n))

• (-p2+r2 :   q2          : p2-r2)
• ( p2           : -q2+r2 : q2-r2)
• (-p2+q2 :   p2-q2 : r2)
• (-1                       : (l2-n2)/(2 m2) :   1)
• ((n2-m2)/(2 l2) : 1                        :  -1)
• ( 1                       : -1                       : (m2-l2)/(2 n2))
DT-Area of QG-P14-Triangle in the QA-environment:
• -(-p2+q2+r2)(p2-q2+r2) (p2+q2-r2) S / (2 p2 q2 r2)
DT-Area of QG-P14-Triangle in the QL-environment:       (equals 3 x area QL-Diagonal Triangle)
• 3 S / 2

Properties:

the line through QG-P1 parallel to QG-L1 (3rd Diagonal).
• QA-P2 (Euler-Poncelet Point) lies on the circumcircle of the triangle formed by the 3 QA-versions of QG-P14.
• QG-P14 lies on the Nine-point Conic QA-Co1. So the vertices of the triangle formed by the 3 QA-versions of QG-P14 lie on the Nine-point Conic QA-Co1.
• QG-P14 is the QA-DT-Trilinear Pole of the QL-DT-Trilinear Polar of QG-P13 (Eckart Schmidt, October 13, 2012). For definition Trilinear Pole and Polar see Ref-13.
• QG-P14 is the 4th point of intersection of the two QA-DT circumscribed conics with centers in the midpoint of the diagonals of the Reference Quadrigon (Eckart Schmidt, November 26, 2012).

Vernieuwen