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 is designated as X in Clark Kimberling's Encyclopedia of Triangle Centers, while the other point called second Morley center is designated as X. 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.

Let be the Morley triangle of. The centroid of is called the first Morley center of.

Let be the Morley triangle of. Then, the lines are concurrent. The point of concurrence is called the second Morley center of triangle.

The trilinear coordinates of the first Morley center of triangle are

The trilinear coordinates of the second Morley center are