Morley centers

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 (or simply, the Morley center ) is designated as X(356) in Clark Kimberling's Encyclopedia of Triangle Centers, while the other point called second Morley center (or the 1st Morley–Taylor–Marr Center ) 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 $△DEF$ be the triangle formed by the intersections of the adjacent angle trisectors of triangle $△ABC$. $△DEF$ is called the Morley triangle of $△ABC$. Morley's trisector theorem states that the Morley triangle of any triangle is always an equilateral triangle.

First Morley center
Let $△DEF$ be the Morley triangle of $△ABC$. The centroid of $△DEF$ is called the first Morley center of $△ABC$.

Second Morley center
Let $△DEF$ be the Morley triangle of $△ABC$. Then, the lines $AD, BE, CF$ are concurrent. The point of concurrence is called the second Morley center of triangle $△ABC$.

First Morley center
The trilinear coordinates of the first Morley center of triangle $△ABC$ are $$\cos \tfrac{A}{3} + 2 \cos \tfrac{B}{3} \cos \tfrac{C}{3} : \cos \tfrac{B}{3} + 2 \cos \tfrac{C}{3} \cos \tfrac{A}{3} : \cos \tfrac{C}{3} + 2 \cos \tfrac{A}{3} \cos \tfrac{B}{3}$$

Second Morley center
The trilinear coordinates of the second Morley center are

$$\sec \tfrac{A}{3} : \sec \tfrac{B}{3} : \sec \tfrac{C}{3}$$