Second Morley Triangle
MorleysSecondTriangle
The second Morley triangle is made by rotating line BC toward vertex A about vertex B by angle (B+2pi)/3. It is an equilateral triangle.
It has trilinear vertex matrix
![画像: [1 2cos[1/3(C-2pi)] 2cos[1/3(B-2pi)]; 2cos[1/3(C-2pi)] 1 2cos[1/3(A-2pi)]; 2cos[1/3(B-2pi)] 2cos[1/3(A-2pi)] 1]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/SecondMorleyTriangle/NumberedEquation1.svg){kind=link}
(Kimberling 1998, p. 165).
Its signed side lengths s^'=a^'=b^'=c^' are
| s^'=8Rsin[1/3(A-2pi)]sin[1/3(B-2pi)]sin[1/3(C-2pi)], |
(2)
|
giving an area of
| A=16sqrt(2)R^2sin^2[1/3(A-2pi)]sin^2[1/3(B-2pi)]sin^2[1/3(C-2pi)]. |
(3)
|
The following table lists perspectors of the second Morley triangles with other named triangles that are Kimberling centers.
triangle Kimberling perspector
first Morley triangle X_(358) second Morley-Taylor-Marr
center
reference
triangle X_(1136) 5th Morley-Taylor-Marr center
third Morley triangle X_(1137) 6th Morley-Taylor-Marr center
See also
First Morley Triangle, Second Morley Adjunct Triangle, Third 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
Second Morley TriangleCite this as:
Weisstein, Eric W. "Second Morley Triangle." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SecondMorleyTriangle.html