QL-27Qu1: Morley’s Multiple Cardioids

It is possible to construct a Cardioid inscribed in a Quadrilateral:


There even are 27 ways to inscribe a Cardioid in a Quadrilateral.
These Cardioids were described in F. Morley’s document Extensions of Cliffords Chain-Theorem. See Ref-37.
He also describes that in a Quadrilateral (4-Line) one single Cardioid will occur enveloping the 4 circumcircles of the Component Triangles (QL-Qu1).
Last but not least he predicts the numbers of other epicycloids occurring in a n-Line.

27 Cardioids inscribed in a Quadrilateral: 
QL-27Qu1-Cardioids-73-27 points

4 Times 3 equilateral triangles in a Quadrilateral are needed to construct them:
QL-27Qu1-Cardioids-71-27 points
The 27 red intersections of the (extended) sides of the 4x3 equilateral triangles are the centers of the inscribed Cardioids. Each center is the intersection point of 4 sidelines.
More information about the subject also can be found in Bernard Keizer’s document at Ref-43.

For the construction of the Cardioids we shall first determine per Cardioid its Center and its Cusp as follows:
  1.    Construct for each Component Triangle its 1st, 2nd and 3rd Morley Triangles. See Ref-13.
This gives per Component Triangle 3 x 3 parallel lines each intersecting at 60°.
Since the Reference Quadrilateral has 4 different Component Triangles the total amount of lines is 4 x 3 x 3. 
These 36 lines intersect somehow in 27 points, where each of these points is intersected by 4 lines. 
These 27 points are the Centers of the 27 Cardioids we are looking for. This property was described by F. Morley in Ref-37.
  2.    Let one of these Centers have barycentric coordinates (u : v : w) wrt a Component Triangle.
Then the cusp has barycentric coordinates (b2 c2 u3 : a2 c2 v3 : a2 b2 w3). 
Found by Eckart Schmidt, see Ref-34, QFG-message # 55. Construct the Cusp by using this property.
  3.    Knowing Cardioid Center P and Cardioid Cusp P1 it is possible to construct the Cardioid:
    1. Let S1 be a variable point on circle Ci(P,P1).
    2. Let S2 be the reflection of P1 in line P.S1.
    3. Let S3 be the reflection of S2 in S1.
    4. The locus of S3 with variable point S1 is the Cardioid. 

Equation QL-27Qu1 in CT-notation:
Equation QL-27Qu1 in DT-notation:

  • The 27 Centers of QL-27Qu1 lie on Eckart’s Cubic QL-Cu2.
  • The centers of Morley’s Multiple Cardioids are called incenters by Morley in Ref-37. There he states that the 27 incenters lie on a network of 36 axes, being the 4x9 Morley axes of the QL-Component Triangles. Each incenter is intersected by 4 axes and each axis contains 3 incenters. See also Ref-34, QFG-messages #1462, #1463.


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