QG-P14: Center of the M3D Hyperbola

QG-P14 Center of the M3D Hyperbola

QG-P14 is the center of the M3D Hyperbola (see paragraph QG-Co3).

M3D stands for “Midpoint 3^rd QA-Diagonal”.

Coordinates:

CT-Coordinates QG-P14 in 3 QA-Quadrigons:

(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))

CT-Coordinates QG-P14 in 3 QL-Quadrigons:

(-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)

DT-Coordinates QG-P14 in 3 QA-Quadrigons:

DT-Coordinates QG-P14 in 3 QL-Quadrigons:

DT-Area of QG-P14-Triangle in the QA-environment:

DT-Area of QG-P14-Triangle in the QL-environment: (equals 3 x area QL-Diagonal Triangle)

Properties:

– QG-P2.QG-P13

– QG-P15.QA-P5

– the line through QG-P1 parallel to QG-L1 (3^rd 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 4^th 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).

MATHEMATICS