Morley centers explained

In plane geometry, the Morley centers are two special points associated with a triangle. Both of them are triangle centers. One of them called first Morley center^[1] (or simply, the Morley center^[2]) is designated as X(356) in Clark Kimberling's Encyclopedia of Triangle Centers, while the other point called second Morley center^[1] (or the 1st Morley–Taylor–Marr Center^[2]) is designated as X(357). The two points are also related to Morley's trisector theorem which was discovered by Frank Morley in around 1899.

Definitions

Let be the triangle formed by the intersections of the adjacent angle trisectors of triangle . is called the Morley triangle of . Morley's trisector theorem states that the Morley triangle of any triangle is always an equilateral triangle.

First Morley center

Let be the Morley triangle of . The centroid of is called the first Morley center of .^{[1] [3]}

Second Morley center

Let be the Morley triangle of . Then, the lines are concurrent. The point of concurrence is called the second Morley center of triangle .^{[1] [3]}

Trilinear coordinates

First Morley center

The trilinear coordinates of the first Morley center of triangle are ^[1] $$\backslash cos\; \backslash tfrac\; +\; 2\; \backslash cos\; \backslash tfrac\; \backslash cos\; \backslash tfrac\; :\; \backslash cos\; \backslash tfrac\; +\; 2\; \backslash cos\; \backslash tfrac\; \backslash cos\; \backslash tfrac\; :\; \backslash cos\; \backslash tfrac\; +\; 2\; \backslash cos\; \backslash tfrac\; \backslash cos\; \backslash tfrac$$

Second Morley center

The trilinear coordinates of the second Morley center are

$$\backslash sec\; \backslash tfrac\; :\; \backslash sec\; \backslash tfrac\; :\; \backslash sec\; \backslash tfrac$$

Notes and References

1. Web site: Kimberling. Clark. 1st and 2nd Morley centers. 16 June 2012.
2. Encyclopedia: Kimberling. Clark. X(356) = Morley center. Encyclopedia of Triangle Centers. 16 June 2012.
3. Web site: Weisstein. Eric W. Morley Centers. Mathworld – A Wolfram Web Resource. 16 June 2012.