QL-P30: Morley’s Second Circle Center


QL-P30 is the center of Morley’s Second Circle as described in Ref-49.
Morley mentions in his paper how in an n-Line (system of n random lines) in a recursive way several centers can be constructed. In his terminology this Second Circle Center is the ratiopoint nL-Center Circle Center : nL-Second Orthocenter ((n-2) : 1).
In EQF-terminology this is QL-P4.QL-P2 (2:1).
Morley’s Second Circle is the circle with center QL-P30 and with radius the QL-Ci3-radius / 3.
The external Homothetic Center of this Second Circle and Miquel Circle QL-Ci3 is QL-P2.
QL-P30 is used for constructing Morley’s Second Orthocenter in a 5-Line.

QL-P30-MorleysSecondCircleCenter-01
 
The Miquel Circle in a Quadrilateral is the equivalent of the circumcircle in a triangle.
Morley’s Second Circle in a Quadrilateral is the equivalent of the Nine-point Circle in a triangle.
 
1st CT-coordinate:
+ a4 (m - n) (-l2 + m n) + b4 (l - 2 m) (l - n) (m - n) + c4 (l - m) (l - 2 n) (m - n)
- b2 c2 (m - n) (2 l2 - 3 l m - 3 l n + 4 m n) + a2 b2 (l - n) (2 m2 + l n - 3 m n) - a2  c2 (l - m) (l m - 3 m n + 2 n2)
 
Properties:
- QL-P2.QL-P4             (1 : 2)
- QL-P3.QL-P20           (2 : 1)
- QL-P5.QL-P29           (1 : 2)
- QL-P12.QL-P27         (-1 : 2 => QL-P30 = Reflection of QL-P27 in QL-P12)
  • QL-P30 is the center of Morley’s Second Circle.
  • QL-P30 is used for the construction of 5L-Morley’s Second Orthocenter.
 



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