**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 Clifford’s 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:*

*4 Times 3 equilateral triangles in a Quadrilateral are needed to construct them:*

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.

*Construction:*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 1

^{st}, 2^{nd}and 3^{rd}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 (b

^{2}c^{2}u^{3}: a^{2}c^{2}v^{3}: a^{2}b^{2}w^{3}).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:

- Let S1 be a variable point on circle Ci(P,P1).
- Let S2 be the reflection of P1 in line P.S1.
- Let S3 be the reflection of S2 in S1.
- The locus of S3 with variable point S1 is the Cardioid.

*Equations:**Equation QL-27Qu1 in CT-notation:*

Pending

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

Pending

*Properties:*- 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.