QG-Ci2: QL-DT-Thales Circle
The QL-DT-Thales Circle is the circle with the line segment between the vertices of the QL-Diagonal Triangle (QG-2P3a and QG-2P3b) that are unequal the Diagonal Crosspoint (QG-P1) as diameter.
Equations:
CT-Equation QG-Ci2 in 1st QA-Quadrigon:
r (-a2 q + b2 q + c2 q + b2 r) x2 - (c p - a r) (c p + a r) y2 - p (b2 p + a2 q + b2 q - c2 q) z2
+ (2 c2 p q - a2 q r + b2 q r - c2 q r + a2 r2 + b2 r2 - c2 r2) x y
+ q (-a2 p + b2 p + c2 p - a2 r - b2 r + c2 r) x z
+ (a2 p2 - b2 p2 - c2 p2 + a2 p q - b2 p q + c2 p q - 2 a2 q r) y z = 0
CT-Equation QG-Ci2 in 1st QL-Quadrigon:
l2 (b m - c n) (b m + c n) x2 + m2 (a2 l2 - a2 l n + b2 l n - c2 l n + c2 n2) y2 + n2 (b m- a l) (b m + a l) z2
+ l m2 (a2 l + b2 l - c2 l - a2 n + b2 n + c2 n) x y + m2 n (a2 l + b2 l - c2 l - a2 n + b2 n + c2 n) y z
+ l n (2 b2 m2 - a2 l n + b2 l n - c2 l n) x z = 0
DT-Equation QG-Ci2 in 1st QA-Quadrigon:
(c2 p2 - a2 r2 + b2 r2) x2 + (2 c2 p2 - a2 r2 + b2 r2 - c2 r2) x y + 2 (c p - a r) (c p + a r) x z +
(a2 p2 - b2 p2 + c2 p2 - 2 a2 r2) y z + (-b2 p2 + c2 p2 - a2 r2) z2 = 0
DT-Equation QG-Ci2 in 1st QL-Quadrigon:
SA x y – SB y2 + b2 x z + SC y z = 0
Properties:
-
These points lie on QG-Ci2:– QG-2P3a/b (intersection points QG-Ci2 with the 3rd QG-Diagonal)– QG-2P5a/b (intersection points QG-Ci2 with the QG-Diagonals)
- QG-Ci2 is orthogonal wrt circles QG-Ci1 and QA-Ci1.
- The intersection points of QG-Ci2 with QG-Ci1 lie on the Perspective Squares Double Cubic QG-2Cu1.
- The intersection points of QG-Ci2 with the diagonals of the Reference Quadrigon (QG-2P5a and QG-2P5b) lie on the cubics QL-Cu1 and QG-2Cu1.