We consider in figure 1 the trisectors of the three angles of the triangle ABC and the intersection points S, T, U of any two of them, which lie nearest to the same side. The three points form an equilateral triangle STU called the MORLEY-triangle.

In figure 2 we see, that the hypocycloid of STEINER of a triangle ABC is in the same way orientated as the MORLEY-triangle of the triangle ABC. With these informations, we can construct the cusps of the hypocycloid.

By trigonometric theorems we get the property for the MORLEY-triangle: |ST|=|TU|=|US|=4*sin(Alpha/3)*sin(Beta/3)*sin(Gamma/3). But there is an other elementary proof too using the theorem of the angle at circumference. See "Mathematische Miniaturen" by H. DÖRRIE, Hirt-Verlag Breslau.