QL-2P2: Isogonal Conjugated Newton Pair of Points


There are exactly 2 points on the Newton Line that are each other's Isogonal Conjugate wrt all 4 QL-Component Triangles.
These points lie on QL-Cu1.
Their midpoint is the intersection point of the Newton Line QL-L1 and the Quasi Ortholine QL-L6. See Ref-34, QFG # 179.
Under certain circumstances QL-2P2a & QL-2P2b can be imaginary points. Then also QL-Cu1 will be bi-partite (divided in 2 curves).
QL-2P2-IsogConj Newton Pair-01


Coordinates:
CT-Coordinates:
(note the + sign in front of the Square Root-part indicates different signs for QL-2P2a & QL-2P2b)
( (l (m - n) + m n) (a2 l2 m2 - b2 l2 m2 + c2 l2 m2 + 2 a2 l2 m n + 2 b2 l2 m n - 2 c2 l2 m n - 2 a2 l m2 n - 2 b2 l m2 n - 2 c2 l m2 n + a2 l2 n2 - b2 l2 n2 + c2 l2 n2 - 2 a2 l m n2 + 2 b2 l m n2 + 2 c2 l m n2 + a2 m2 n2 - b2 m2 n2 + c2 m2 n2
+ (-4 b2 c2 (l2 (m - n)2 - m2 n2)2 + (b2 (l (m - n) + m n)2 + c2 (m n + l (-m + n))2 - a2 (m n - l (m + n))2)2)) :
-(-m n + l (m + n)) (a2 l2 m2 - b2 l2 m2 - c2 l2 m2 + 2 a2 l2 m n + 2 b2 l2 m n + 2 c2 l2 m n - 2 a2 l m2 n - 2 b2 l m2 n + 2 c2 l m2 n + a2 l2 n2 - b2 l2 n2 - c2 l2 n2 - 2 a2 l m n2 + 2 b2 l m n2 - 2 c2 l m n2 + a2 m2 n2 - b2 m2 n2 - c2 m2 n2
+ (-4 b2 c2 (l2 (m - n)2 - m2 n2)2 + (b2 (l (m - n) + m n)2 + c2 (m n + l (-m + n))2 - a2 (m n - l (m + n))2)2)) :
2 c2 (l (m - n) - m n) (l (m - n) + m n) (-m n + l (m + n)) )

Properties:
  • QL-2P2a & QL-2P2b lie on QL-L1, QL-Cu1.
  • QL-2P2a & QL-2P2b are mutual QL-Tf1 images as well as QG-Tf2 images (Ref-34, Eckart Schmidt in EQF message #217).
  • The midpoint of QL-2P2a & QL-2P2b is the intersection point QL-L1 ^ QL-L6.
  • The circle through QL-2P2a & QL-2P2b and QL-P1 is the QL-Tf1 image of the Newton Line QL-L1 and is tangent at QL-Cu1 in QL-P1.

 

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