QL-Qu1: Morley's Mono Cardioid


QL-Qu1 is the Cardioid which is the envelope of the circles through fixed point QL-P1 (Miquel Point) and with circumcenter on QL-Ci3 (Miquel Circle).
The equation is of the 4th degree, so it is a quartic.
QL-Qu1 was described by F. Morley in his document “Extensions of Clifford’s Chain-Theorem”. See Ref-37. There he describes that in a Quadrilateral (4-Line) one single Cardioid will occur enveloping the 4 circumcircles of the Component. Also he describes that there will be 27 possible Cardioids (QL-27Qu1) that can be inscribed in a Quadrilateral (4-Line). He describes also the numbers of other epicycloids occurring in a n-Line.
QL-Qu1 is described by Eckart Schmidt (see Ref-15d) and by Bernard Keizer in his document at Ref-43 as well as in Ref-34, QFG #514, #918.
QL-Qu1-01
 
Construction-methods:
  1. QL-Qu1 is the locus of the reflections of QL-P1 in tangents at QL-Ci3 (QFG#918).
  2. QL-Qu1 is the QL-Tf1 image of the inscribed parabola QL-Co1 (QFG#918).
  3. Consider a circle through QL-P1 and centered in the reflection of QL-P1 in QL-P4. The pedal points of QL-P1 wrt tangents at this circle give the cardioid (QFG#918).
  4. QL-Cu1 is the Catacaustic (see Ref-13) of  a circle round QL-P4 through the ratiopoint QL-P1.QL-P4 (4:-3). Rays from this point envelop with their reflections at the circle the cardioid (QFG#918).
  5. Another way of constructing the mono-Cardioid is passingly described by F. Morley at Ref-47, page 20. Take 2 circumscribed circles of Component Triangles in the Quadrilateral. They are tangent to the Cardioid in 2 points: the Miquel point QL-P1 and one vertice of the Quadrilateral. Take a variable line through this vertice, it cuts the 2 circles in 2 points and the tangents to the 2 circles in these points intersect on the Cardioid (see also Ref-34, QFG-message #811 of Bernard Keizer).


Equation/Coordinates:     
Equation in CT-notation:
   a4 (m - n)2 Ta2  + b4 (l - n)2 Tb2  + c4 (l - m)2 Tc2
+ 2 a2 b2 (n - l) (n - m) Ta Tb + 2 b2 c2 (l - m) (l - n) Tb Tc + 2 a2 c2 (m - l) (m - n) Tc Ta = 0
where:
Ta = a2 ( l - m) ( l - n) y z + c2 ( l - n) y (l x + m y + m z) + b2 (l - m) z ( l x + n y + n z)
Tb = b2 (m - l) (m - n) x z + c2 (m - n) x (l x + m y + l z) + a2 (m - l) z (n x + m y + n z)
Tc = c2 (n - m) (n - l) x y + b2 (n - m) x (l x + l y + n z) + a2 ( n - l) y (m x + m y + n z)
Equation in DT-notation:
                (l2 - n2) (m2 - n2) (-2 a2 b2 n (m x + l y) z + a4 m n z (x - y + z) + b4 l n z (-x + y + z) -
c4 l m z (x + y + z) - 2 b2 c2 l (-l2 x2 + (y + z) (m2 y + l2 z)) + 2 a2 c2 m (-m2 y2 + (x + z) (l2 x + m2 z)))2
+ (l2 - m2) (l2 - n2) (b4 l n x (x + y - z) + c4 l m x (x - y + z) - a4 m n x (x + y + z) - 2 b2 c2 l x (n y + m z) + 2 a2 b2 n ((x + y) (n2 x + m2 y) - n2 z2) - 2 a2 c2 m (-m2 y2 + (x + z) (m2 x + n2 z)))2
- (l2 - m2) (m2 - n2) (a4 m n y (x + y - z) + c4 l m y (-x + y + z) - b4 l n y (x + y + z)
- 2 a2 c2 m y (n x + l z) - 2 a2 b2 n ((x + y) (l2 x + n2 y) - n2 z2) + 2 b2 c2 l (-l2 x2 + (y + z) (l2 y + n2 z)))2 = 0 


Properties:
  • The cusp of the Cardioid is at QL-P1, the Miquel Point.
  • The inner circle of the Cardioid is QL-Ci3 (Miquel Circle).
  • QL-Qu1 is the Clawson-Schmidt conjugate of the inscribed QL-Parabola QL-Co1.
  • The angle between the axis of the Cardioid and the axis of the inscribed QL-Parabola QL-Co1 is ∑θi, where θi represents the angles between the Steiner Line and the lines Li (i=1,2,3,4) of the Reference Quadrilateral (Bernard Keizer, April 17, 2013).
  • QL-Qu1 is tangent to the 4 circles circumscribing the 4 Component Triangles of the Reference Quadrilateral (Bernard Keizer, April 17, 2013).

 

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