The third Morley triangle is made by rotating line 
toward vertex 
about vertex 
by angle 
. It is an equilateral
triangle.
It has trilinear vertex matrix
![[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]](/images/equations/ThirdMorleyTriangle/NumberedEquation1.svg) |
(1)
|
(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)]](/images/equations/ThirdMorleyTriangle/NumberedEquation2.svg) |
(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)].](/images/equations/ThirdMorleyTriangle/NumberedEquation3.svg) |
(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 |  | 4th Morley-Taylor-Marr center |
| reference triangle |  | third Morley-Taylor-Marr center |
| second Morley triangle |  | 6th Morley-Taylor-Marr center |
