QLQu1: Morley's Mono Cardioid
QLQu1 is the Cardioid which is the envelope of the circles through fixed point QLP1 (Miquel Point) and with circumcenter on QLCi3 (Miquel Circle).
The equation is of the 4^{th} degree, so it is a quartic.
QLQu1 was described by F. Morley in his document “Extensions of Clifford’s ChainTheorem”. See Ref37. There he describes that in a Quadrilateral (4Line) one single Cardioid will occur enveloping the 4 circumcircles of the Component. Also he describes that there will be 27 possible Cardioids (QL27Qu1) that can be inscribed in a Quadrilateral (4Line). He describes also the numbers of other epicycloids occurring in a nLine.
QLQu1 is described by Eckart Schmidt (see Ref15d) and by Bernard Keizer in his document at Ref43 as well as in Ref34, QFG #514, #918.
Constructionmethods:
 QLQu1 is the locus of the reflections of QLP1 in tangents at QLCi3 (QFG#918).
 QLQu1 is the QLTf1 image of the inscribed parabola QLCo1 (QFG#918).
 Consider a circle through QLP1 and centered in the reflection of QLP1 in QLP4. The pedal points of QLP1 wrt tangents at this circle give the cardioid (QFG#918).
 QLCu1 is the Catacaustic (see Ref13) of a circle round QLP4 through the ratiopoint QLP1.QLP4 (4:3). Rays from this point envelop with their reflections at the circle the cardioid (QFG#918).
 Another way of constructing the monoCardioid is passingly described by F. Morley at Ref47, page 20. Take 2 circumscribed circles of Component Triangles in the Quadrilateral. They are tangent to the Cardioid in 2 points: the Miquel point QLP1 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 Ref34, QFGmessage #811 of Bernard Keizer).
Equation/Coordinates:
Equation in CTnotation:
a^{4} (m  n)^{2} T_{a}^{2} + b^{4} (l  n)^{2} T_{b}^{2} + c^{4} (l  m)^{2} T_{c}^{2}
+ 2 a^{2} b^{2} (n  l) (n  m) T_{a} T_{b} + 2 b^{2} c^{2} (l  m) (l  n) T_{b} T_{c} + 2 a^{2} c^{2} (m  l) (m  n) T_{c} T_{a} = 0
where:
T_{a} = a^{2} ( l  m) ( l  n) y z + c^{2} ( l  n) y (l x + m y + m z) + b^{2} (l  m) z ( l x + n y + n z)
T_{b} = b^{2 }(m  l) (m  n) x z + c^{2} (m  n) x (l x + m y + l z) + a^{2} (m  l) z (n x + m y + n z)
T_{c} = c^{2} (n  m) (n  l) x y + b^{2} (n  m) x (l x + l y + n z) + a^{2} ( n  l) y (m x + m y + n z)
Equation in DTnotation:
(l^{2}  n^{2}) (m^{2}  n^{2}) (2 a^{2} b^{2} n (m x + l y) z + a^{4} m n z (x  y + z) + b^{4} l n z (x + y + z) 
c^{4} l m z (x + y + z)  2 b^{2} c^{2} l (l^{2} x^{2} + (y + z) (m^{2} y + l^{2} z)) + 2 a^{2} c^{2} m (m^{2} y^{2} + (x + z) (l^{2} x + m^{2} z)))^{2}
+ (l^{2}  m^{2}) (l^{2}  n^{2}) (b^{4} l n x (x + y  z) + c^{4} l m x (x  y + z)  a^{4} m n x (x + y + z)  2 b^{2} c^{2} l x (n y + m z) + 2 a^{2} b^{2} n ((x + y) (n^{2} x + m^{2} y)  n^{2} z^{2})  2 a^{2} c^{2} m (m^{2} y^{2} + (x + z) (m^{2} x + n^{2} z)))^{2}
 (l^{2}  m^{2}) (m^{2}  n^{2}) (a^{4} m n y (x + y  z) + c^{4} l m y (x + y + z)  b^{4} l n y (x + y + z)
 2 a^{2} c^{2} m y (n x + l z)  2 a^{2} b^{2} n ((x + y) (l^{2} x + n^{2} y)  n^{2} z^{2}) + 2 b^{2} c^{2} l (l^{2} x^{2} + (y + z) (l^{2} y + n^{2} z)))^{2} = 0
Properties:

The cusp of the Cardioid is at QLP1, the Miquel Point.

The inner circle of the Cardioid is QLCi3 (Miquel Circle).

The angle between the axis of the Cardioid and the axis of the inscribed QLParabola QLCo1 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).

QLQu1 is tangent to the 4 circles circumscribing the 4 Component Triangles of the Reference Quadrilateral (Bernard Keizer, April 17, 2013).