QA-Tf12: QA-Pedal Point

The QA-Pedal Point is defined as follows.
• The 3 Pedal Points of P wrt a Triangle define a Pedal Circle Ci.
• In a Quadrangle the 4 Pedal Circles Ci (denoted as Ci1, Ci2, Ci3, Ci4) of the Component Triangles intersect at a common point being the QA-Pedal Point.
For more information and background on this transformation see Ref-34, QFG-messages #1218, #1219, #1222, #1223, #1231, #1232, #2759.
 QA Tf12 QA Pedal Point 01
1st CT-coordinate of QA-Tf12[(x:y:z)]:
p (r y - q z)
(c2 (2 b2 p + a2 q + b2 q - c2 q) r x3 y + c2 (a2 p + b2 p - c2 p + 2 a2 q) r x2 y2 + b2 q (-2 c2 p - a2 r + b2 r - c2 r) x3 z - 2 p (a2 c2 q + b2 c2 q - c4 q - a2 b2 r + b4 r - b2 c2 r) x2 y z + (-2 a2 c2 p q + a4 p r - b4 p r + 2 b2 c2 p r - c4 p r + a4 q r - a2 b2 q r + a2 c2 q r) x y2 z - b2 q (a2 p - b2 p + c2 p + 2 a2 r) x2 z2 + (-a4 p q + b4 p q - 2 b2 c2 p q + c4 p q + 2 a2 b2 p r - a4 q r - a2 b2 q r + a2 c2 q r) x y z2 - a2 p (a2 q - b2 q + c2 q - a2 r - b2 r + c2 r) y2 z2)


• Every point on a side of a Quadrangle is transformed in the projection point on the opposite side of the Quadrangle.
QA-Tf12 has the vertices of the Diagonal Triangle as fixed poins. See Ref-34, QFG#1231.
QA-P4 is transformed into QA-P2. See Ref-34, QFG#1219.
• Not only QA-P4 is transformed into QA-P2, but all points of the QA-Orthogonal Hyperbola QA-Co2. See Ref-34, QFG#1231.
QA-Tf12(P) = CO-Tf3-1(P) wrt the conic(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).
QA-P12 can be generalized as the transformation nP-e-Tf1 in an n-Point (system of n reference points). It matches with nP-e-Tf1 when n=4. See Ref-34, QFG#2759.
• 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.
• The locus of QA-Tf6(L) wrt a pencil of lines L through a random point P is a circle. It is the same circle as occurs in the construction of QA-Tf3. Its center is QA-Tf3(P). QA-P2 and QA-Tf12(P) lie on this circle. The point QA-Tf12(P) is identical with 5P-s-Tf3(P) of Pentangle P1.P2.P3.P4.P. The angle deviation of L through P is transformed in a doubled angle deviation of QA-Tf6(P) on this circle in the opposite direction.

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