The other way around...

And the other way around? Can all equilateral triangles be obtained by the intersection of adjacent angle trisectors of a certain triangle? In fact this is always possible, actually there are an infinite number of triangles that give origin to the same equilateral triangle. This can be observed with the following applet, where the triangle \([ABC]\) is the initial triangle and the equilateral triangle \([DEF]\) is its Morley’s triangle

(Click on the point \(A\) and move it inside the area delimited by the red half-lines to obtain several solutions. You can also click on the vertices \(E\) and \(F\) of the equilateral triangle and move them in order to obtain another equilateral triangle, without forgetting to move the point \(A\) after so that it stays in the half-line delimited area)