The following theorem is unusual because, despite its simplicity, it was only invented about 100 years ago. According to H.S.M. Coxeter, this may be because it involves angle trisectors, which cannot in general be constructed with ruler and straightedge.
The theorem is simple: trisect the angles of a triangle. Find three points where the trisectors meet. These points will define an equilateral triangle.
The following illustration shows Morley's triangle in green. Try dragging the large triangle's red vertices with the mouse (if your mouse has more than one button, use the left one).
David E. Joyce of Clark University has written a much better system that specializes in demonstrating geometric constructions. When I saw his system, I realized I could adapt mine to achieve a similar effect. His Geometry applet is available here.