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First Morley Triangle


MorleysTriangle

The first Morley triangle DeltaA^'B^'C^', also simply known as "Morley's triangle", is the triangle constructed from pairwise intersections of the angle trisectors of a given triangle DeltaABC. In particular, its A^'-vertex is the point of intersection of the first trisector of B when swinging BC counterclockwise about B and the first trisector of C, with the B^'- and C^'-vertices similarly defined (Kimberling 1998, p. 165). By Morley's theorem, this triangle is equilateral.

It has trilinear vertex matrix

 [1 2cos(1/3C) 2cos(1/3B); 2cos(1/3C) 1 2cos(1/3A); 2cos(1/3B) 2cos(1/3A) 1]

(Kimberling 1998, p. 165).

It has side lengths

 a^'=b^'=c^'=8Rsin(1/3A)sin(1/3B)sin(1/3C),

where R is the circumradius of the original triangle.

All triangle centers of the first Morley triangle concur at the first Morley center of the reference triangle.

The following table lists perspectors of the second Morley triangles with other named triangles that are Kimberling centers.

excentral triangleX_(1507)first Morley-Gibert point
first Morley adjunct triangleX_(356)first Morley center
reference triangleX_(357)first Morley-Taylor-Marr center
second Morley triangleX_(358)second Morley-Taylor-Marr center
third Morley triangleX_(1135)4th Morley-Taylor-Marr center

See also

First Morley Adjunct Triangle, Marion's Theorem, Morley's Theorem, Second Morley Triangle, Third Morley Triangle

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References

Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.

Referenced on Wolfram|Alpha

First Morley Triangle

Cite this as:

Weisstein, Eric W. "First Morley Triangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FirstMorleyTriangle.html

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