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 [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 4×3 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 [43].

Construction

For the construction of the Cardioids we shall first determine per Cardioid its Center and its Cusp as follows:

• Construct for each Component Triangle its 1^st, 2^nd and 3^rd Morley Triangles. See [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 [37].
• Let one of these Centers have barycentric coordinates (u : v : w) wrt one of the Component Triangles. Then the cusp has barycentric coordinates (b² c² u³ : a² c² v³ : a² b² w³). Found by Eckart Schmidt, see [34], QFG-message # 55. Construct the Cusp by using this property.
• Knowing the Cardioid Center P and the 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.

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 [37]. There he states that the 27 incenters lie on a network of 36 axes, being the 4×9 Morley axes of the QL-Component Triangles. Each incenter is intersected by 4 axes and each axis contains 3 incenters. See also [34], QFG-messages #1462, #1463.

Estimated human page views: 663