The aims of this research are to study the internal and external Morley triangles whose mother triangles are (1) equilateral triangles, (2) right triangles, and (3) isosceles triangles.

So far, we have investigated the external Morley triangle of our first mother triangle, an equilateral triangle. By solving systems of linear equations with trigonometric functions, we are able to find the coordinates of the three vertices of the external Morley and how they are related to the mother triangles. (See below) Moreover, parallelism and collinearity of vertices and centroids are also studied.

Given the mother triangle ABC with A= (-a, 0), B = (a, 0), and C=(0,c), we can obtain the following figure and find the coordinates of the vertices of the external Morley triangle PQR as follows.