QA-P16

QA-P16: QA-Harmonic Center

The triangle formed by the 3 QA-versions of QG-P12 (Inscribed Harmonic Conic Center) is perspective with the QA-Diagonal Triangle. Their Perspector is QA-P16.

QA-P16 partakes in many QA-Parallelities and with many QA-Crosspoints (see QA-3 and QA-5).

QA-P16 is named harmonic because its construction is based on projective principles leading to harmonic properties.

QA-P16 in a Quadrangle is the counterpart of QL-P13 in a Quadrilateral.

Construction:

Construct QA-P16 as the Complement of the Isotomic Conjugate of the AntiComplement of QA-P1 wrt the QA-Diagonal Triangle

= QA-P10-Ceva conjugate of QA-P1 wrt the QA-Diagonal Triangle .

QA-P16 is the perspector of the triangles formed by the 3 QA-versions of QG-P12 (Inscribed Harmonic Conic Center) and the QA-Diagonal Triangle.

QA-P16 is the intersection point of the tangents at the vertices of the QA-Diagonal Triangle and QA-P10 to the QA-DT-P10 Cubic (QA-Cu3).

Coordinates:

1^st CT-coordinate:

p (2 p + q + r)

1^st DT-coordinate:

p²

Properties:

QA-P16 lies on these QA-lines:

– QA-P1.QA-P21 (-1 : 2 => QA-P16 = Reflection QA-P21 in QA-P1)

– QA-P10.QA-P19 (-1 : 3)

QA-P16 lies on this QG-line:

– QG-P1.QG-P12 = QG-L2

QA-P16 is the Reflection of QA-P19 in QA-P31.
QA-P16 is the QA-Involutary Conjugate (see QA-Tf2) of QA-P10.
QA-P16 = QA-P10-Ceva conjugate of QA-P1 wrt the QA-Diagonal Triangle.
QA-P16 is collinear with QG-P1, QG-P12, QG-P13, QL-P13 on QG-L2.
QA-P16 is the 4^th Perspective Point in the row QG-P13, QG-P12, QG-P1 on line QG-L2 (see Ref-26 Perspective Fields part II).
QA-P16 lies on the Conic QA-Co5.
QA-P16 lies on the Conic through P1, P2, P3, P4 and QA-P1.
QA-P16 lies on the Cubics QA-Cu3 and QA-Cu4.
QA-P16 is the pole of the Cubics QA-Cu1 – QA-Cu5 when seen as IsoCubics wrt the QA-Diagonal Triangle and with the Involutary Conjugate as Isoconjugation.
QA-P16 is the intersection point of the tangents at the vertices of the QA-Diagonal Triangle and QA-P10 to the QA-DT-P10 Cubic (QA-Cu3).
QA-P16 is the Complement of QA-P19 wrt the QA-Diagonal Triangle.
QA-P16 is the AntiComplement of QA-P31 wrt the QA-Diagonal Triangle.
The 3 versions of QG-L2 in a Quadrangle concur in QA-P16.
QA-P16 is the perspector the QA-Diagonal Triangle and the tangential triangle of the QA-Nine-point conic wrt the QA-Diagonal Triangle (note Randy Hutson).
QA-P16 is the Radical Center of the 3 QA-versions of QL-Ci1: Circumcircle QL-DT (Eckart Schmidt, July, 2012).
Let P1P2P3P4 be a Quadrangle. Let Qi (i=1,2,3,4) be the center of the circum-conic to Diagonal Triangle with perspector Pi. QA-P16 is the common intersection point of lines Pi.Qi (Angel Montesdeoca, January 18, 2015).
QA-P16 is the Perspector of the mutual Triple Triangles (see QA-Tr-1) of QG-P1, QG-P12, QG-P13, QL-P13.

MATHEMATICS

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