QA-P41: Involutary Conjugate of QA-P4

QA-P41 is the Involutary Conjugate of QA-P4 (Isogonal Center).

It is also a special point because it is the common point of the 3 QA-versions of QG-Ci3 (Quasi Isogonal Circumcenter).

1st CT-Coordinate

a² q r (a² p² – b² p² – c² p² + a² p q – b² p q + a² p r – c² p r + a² q r)

(-c⁴ p³ q³ + a² c² p³ q² r – c⁴ p³ q² r + b² c² p² q³ r – c⁴ p² q³ r + a² b² p³ q r² – b⁴ p³ q r² + a⁴ p² q² r² – b⁴ p² q² r² + 2 b² c² p² q² r² – c⁴ p² q² r² + a⁴ p q³ r² – a² b² p q³ r² – b⁴ p³ r³ – b⁴ p² q r³ + b² c² p² q r³ + a⁴ p q² r³ – a² c² p q² r³ + a⁴ q³ r³)

1st DT-Coordinate

p² / (-b² c² p⁴ + b² c² p² q² – c⁴ p² q² – b⁴ p² r² + b² c² p² r² + a⁴ q² r²)

Properties

• QA-P41 is the common intersection point of the 3 QA-versions of QG-Ci3 (Quasi Isogonal Circumcircle).
• QA-P41 lies on the circumcircle of the triangle formed by the 3 QA-versions of QG-P18.
• QA-P41.QA-P11 // QA-P2.QA-P23
• QA-P41 lies on the cubics QA-Cu1 and QA-Cu7.
• The QA-Möbius Conjugate (QA-Tf4) of QA-P41 is the intersection point QA-P2.QA-P4 ^ QA-L3.QA-P32.
• QA-circumconics intersect QA-Cu1 in two further points collinear with QA-P41. See [34], Eckart Schmidt, QFG-message #1666.
• The tangent at QA-P4 and the tangents at the vertices of the Diagonal Triangle (QA-Tr1) to QA-Cu1 are concurrent in QA-P41.
• Let QG-P1a, QG-P1b, QG-P1c be the three QA-versions of QG-P1 and let QG-P18a, QG-P18b, QG-P18c be the three QA-versions of QG-P18. QA-P41 is the common point of the circles (QG-P18a, QGP1b, QGP1c), (QG-P18b, QGP1c, QGP1a) and (QG-P18c, QGP1a, QGP1b).

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