X₃₅₆

MORLEY CENTER

X₃₅₆ is the centroid of the Morley equilateral triangle. For a discussion of the theorem and extensive list of references, see

5, 6, 7 3, 4, 5 Kepler Kimberling

X₃₅₆ (MORLEY CENTER) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {357, 358}, {1134, 1135}

• Related Centers: X₃₆₀₅ (ISOGONAL CONJUGATE OF X₃₅₆), X₃₅₇ (1st MORLEY-TAYLOR-MARR CENTER), X₃₅₈ (2nd MORLEY-TAYLOR-MARR CENTER), X₁₁₃₄ (3rd MORLEY-TAYLOR-MARR CENTER), X₁₁₃₅ (4th MORLEY-TAYLOR-MARR CENTER)

Trilinears: \(\cos A/3 + 2 \cos B/3 \cos C/3 : :\)

Search Value (6, 9, 13): \(1.625386564358\)

▸ Alternate Coordinates1 form

• Trilinears: \(\cos(B/3 - C/3) + \sqrt{3}\sin(A/3 + \pi /3)\) (M. Stevanovic, 12/25/2007)

▸ Linear Combinations1 combination

• X₃₅₆ = X₃₂₇₆ + X₃₂₇₇ - X₃

▸ Incident Central Lines5 lines

{2, 5455}, {3, 3278}, {357, 358}, {1134, 1135}, {3605, 5456}

▸ Functional Relations2 relations

• X₃₅₆ = isogonal conjugate of X₃₆₀₅

• X₃₅₆ = orthology center of 1st Morley adjunct triangle and 1st Morley triangle

Notes

C. O. Oakley and J. C. Baker, “The Morley trisector theorem,” American Mathematical Monthly 85 (1978) 737-745.

For a sketch of the Morley cubic and list of centers on it, including X₃₅₆, X₃₅₇, X₃₅₈, see Bernard Gibert’s site.

See also Dao Thanh Oai, Some new equilateral triangles in a plane geometry.

Let A'B'C' be the 1st Morley triangle. Let A″ be the cevapoint of B' and C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₃₅₆. (Randy Hutson, September 14, 2016)

For a biographical sketch, including details about Morley’s famous theorem on angle trisectors, with history and references, see

Frank Morley (1860-1937) geometer (../bstud/morley.html).