QA-Tf14 is a point to point transformation wrt a Quadrangle.
It is based upon the Orthotransversal transformation in a triangle.
It is based upon the Orthotransversal transformation in a triangle.
Definition Orthotransversal Line wrt a Triangle
Let ABC be the Reference Triangle and let P be a random point. The perpendiculars at P to AP, BP and CP respectively meet BC, AC and AB in three collinear points defining a line which is being called the orthotransversal (line) of P.
CT-Coordinates of QA-Tf14[(x:y:z)]:
(-2 a2 p r y2 z + 2 a2 p q y z2 + x2 (-p (a2 r - b2 r - c2 r) y - p (-a2 q + b2 q + c2 q) z) + x (-p (a2 r - b2 r + c2 r) y2 - p (-a2 q + b2 q - c2 q + a2 r + b2 r - c2 r) y z - p (-a2 q - b2 q + c2 q) z2) :
(a2 p - b2 p + c2 p) q y2 z + (-a2 p - b2 p + c2 p) q y z2 + x2 (q (-a2 r + b2 r + c2 r) y + 2 b2 q r z) + x (q (-a2 r + b2 r - c2 r) y2 + q (a2 p - b2 p - c2 p + a2 r + b2 r - c2 r) y z - 2 b2 p q z2) :
-(-a2 p + b2 p - c2 p) r y2 z - (a2 p + b2 p - c2 p) r y z2 + x2 (-2 c2 q r y - (-a2 q + b2 q + c2 q) r z) + x (2 c2 p r y2 - (a2 p - b2 p - c2 p + a2 q - b2 q + c2 q) r y z - (-a2 q - b2 q + c2 q) r z2))
Properties:
• QA-Tf14(QA-P2) = QA-P4.
• QA-Tf14(QA-P23) is a point on QA-L9.
• QA-Tf14(P) = CO-Tf3(P) wrt the conic defined by P1,P2,P3,P4,P, where P1,P2,P3,P4 are the vertices of the Reference Quadrangle (personal mail Benedetto Scimemi and see Ref-34, QFG#1252).
• Consequently in a 5-Point configuration all versions of QA-Tf14(P) wrt the 5 Component Quadrangles coincide with CO-Tf3(P) when P is a point on the circumscribed conic of the 5-Point (5P-s-Co1), like described by Bendetto Scimemi in a personal mail and at Ref-34, QFG#1252.
• Given a Pentangle {P1, P2, P3, P4, P5}.
Let A1 = QA-Tf14 (P1) wrt {P2,P3,P4,P5}; similarly we define A2, A3, A4, A5.
Let B1 = QA-Tf12 (P1) wrt {P2,P3,P4,P5}; similarly we define B2, B3, B4, B5.
Then two pentangle {P1,P2,P3,P4,P5}, {A1,A2,A3,A4,A5} are inversely similar and the two pentangles {A1,A2,A3,A4,A5}, {B1, B2, B3, B4. B5} are homothetic with the center of the circumconic of {P1,P2,P3,P4,P5} as homothetic center. The circumconics of {P1,P2,P3,P4,P5}, {A1,A2,A3,A4,A5} and {B1, B2, B3, B4. B5} are similar and share the same center. See Vu Thunh Tang at Ref-59b.
• In a Pentangle the 5 versions of QA-Tf14(P) define a 5P-Conic. In a Hexangle (6-Point) the 6 versions of these 5P-Conics form a pencil of conics with 3 common points.
• In a Pentangle the 5 versions of QA-Tf14(P) define a 5P-Conic. In a Hexangle (6-Point) the 6 versions of these 5P-Conics form a pencil of conics with 3 common points.