QG-P19

QG-P19: Quasi Isogonal Conjugate of QG-P1

QG-P19 is the intersection of the reflections of the two QA-DT-side lines unequal QG-L1 in the angle bisectors of the opposite sides of the quadrigon.

This point and its properties are found by Eckart Schmidt (December 12, 2012).

QG-P19-Quasi Isogonal Conjugate QG-P1-01

Coordinates:

CT-Coordinates in 1^st QA-Quadrigon:

(a² q r ((p + q) (b² p + a² q) r² + c² p q (p q - r²)) : -p r (b⁴ p (p + q) r (q + r) - (a² - c²) q² (a² (p + q) r - c² p (q + r)) + b² q (-a² (p + q) (p - q - r) r + c² p (p + q - r) (q + r))) : c² p q (c² p² q (q + r) + r (b² p² (q + r) + a² q (-p² + q r))))

DT-Coordinates in 1^st QA-Quadrigon:

(-(a² + b² - c²) p² (c² q² + b² r²) : (b² p² + a² q²) (c² q² + b² r²) : (a² - b² - c²) (b² p² + a² q²) r²)

CT-Coordinates in 1^st QL-Quadrigon:

(a² l (-a² (l - m) (l - n) + c² (l - n) (-m + n) + b² (l (m - n) + m (-2 m + n))) : b² m (a² l² + c² n²) : c² n (-a² (l - m) (l - n) + c² (l - n) (-m + n) + b² (l (m - n) + m (-2 m + n))))

DT-Coordinates in 1^st QL-Quadrigon:

(-m² (a⁴ (l² - n²) + b² (b² - c²) (m² - n²) + a² (c² (-l² + n²) + b² (l² + m² + 2 n²))) : 2 b² c² (l² + m²) n² + c⁴ n² (-l² + n²) + 2 a² b² l² (m² + n²) + a⁴ (l⁴ - l² n²) + b⁴ (m⁴ - l² n²) : m² (b⁴ (l² - m²) + c⁴ (l² - n²) - b² c² (2 l² + m² + n²) + a² (b² (-l² + m²) + c² (-l² + n²))))

Properties:

The QL-Tf1 image of QG-P19 is a point on QA-Ci1.
QG-P19 is the QG-Tf2 image of QG-P1.
QG-P19 is the QA-Tf2 image of the QG-P18.
Let S_i = QG-P1, X_i = QG-P18 and Y_i = QG-P19 resp. in QA-Quadrigon-i (i=1,2,3).

Now S_i.X_j ^ S_j.X_i = Y_k, S_i.Y_j ^ S_j.Y_i = X_k, X_i.Y_j ^ X_j.Y_i = S_k.

The Triangles formed by the QA-versions of QG-P1, QG-P18 and QG-P19 are pairwise perspective triangles.

Their centers of perspectivity are collinear.

QA-P4 lies on the circle defined by the 3 QA-versions of QG-P19.
QG-P19 is concyclic with QG-2P2a, QG-2P2b, QA-P4 (the circle is the QL-Tf1 image of QA-Ci1).
QG-P19 lies on QA-Cu7 (as 4^th intersection with the circle through QG-2P2a, QG-2P2b, QA-P4).
The altitudes of the Triangle formed by the 3 QA-versions of QG-P19 are concurrent in a point on QA-P12.QA-P24.
The Polar (see Ref-13, Polar) of QG-P19 wrt any circumscribed conic of the Reference Quadrigon is a line through QG-P18.
The Triple Triangle of QG-P19 is perspective with all QA-Component Triangles (see QA-Tr-1 for Desmic Triple Triangles).

MATHEMATICS

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