Let:

Q lie on AC and N lie outside http://www.cut-the-knot.org/gifs/triangle.gif ABC;
http://www.cut-the-knot.org/gifs/triangle.gif AQN be congruent to http://www.cut-the-knot.org/gifs/triangle.gif BPM;
D be the centre of http://www.cut-the-knot.org/gifs/triangle.gif AQN.

Then:

DF passes through E (the centre of symmetry between triangles AQN and PMB);
http://www.cut-the-knot.org/gifs/triangle.gif BCD is congruent to http://www.cut-the-knot.org/gifs/triangle.gif ACF.

So:

http://www.cut-the-knot.org/gifs/triangle.gif DCF is equilateral and CE is its height.

http://geocities.com/narbab/ctk/equilat.png