Third Morley Triangle
MorleysThirdTriangle
The third Morley triangle is made by rotating line BC toward vertex A about vertex B by angle (B+4pi)/3. It is an equilateral triangle.
It has trilinear vertex matrix
![画像: [1 2cos[1/3(C-4pi)] 2cos[1/3(B-4pi)]; 2cos[1/3(C-4pi)] 1 2cos[1/3(A-4pi)]; 2cos[1/3(B-4pi)] 2cos[1/3(A-4pi)] 1]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/ThirdMorleyTriangle/NumberedEquation1.svg){kind=link}
(Kimberling 1998, p. 165).
Its signed side lengths are
| a^'=b^'=c^'=8Rsin[1/3(A-4pi)]sin[1/3(B-4pi)]sin[1/3(C-4pi)] |
(2)
|
giving an area of
| A=16sqrt(2)R^2sin^2[1/3(A-4pi)]sin^2[1/3(B-4pi)]sin^2[1/3(C-4pi)]. |
(3)
|
The following table lists perspectors of the third Morley triangles with other named triangles that are Kimberling centers.
triangle Kimberling perspector
first Morley triangle X_(1135) 4th Morley-Taylor-Marr
center
reference
triangle X_(1134) third Morley-Taylor-Marr center
second Morley triangle X_(1137) 6th Morley-Taylor-Marr center
See also
First Morley Triangle, Second Morley TriangleExplore with Wolfram|Alpha
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References
Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Referenced on Wolfram|Alpha
Third Morley TriangleCite this as:
Weisstein, Eric W. "Third Morley Triangle." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ThirdMorleyTriangle.html