Draw an equilateral triangle. Divide it into four identical shapes. Remove the bottom left-hand shape. Now divide the resulting shape into four identical shapes.

Orient the triangle as if it was a pyramid. Connect the midpoints of the three sides, to form 4 triangles. Remove the bottom left-hand triangle, leaving an isosceles trapezoid (which I am informed is a trapezium outside the US, and a non-trapezium inside the US.).

Now, there are three triangles, with four line segments meeting at a point. (I hope that you haven't erased the lines yet.) Bisect each of those four line segments, and join them (without crossing any lines). Each of the three triangles is now divided into an isosceles trapezoid and a smaller triangle, and those three smaller triangles constitute the 4th, similar trapezoid.

p.s. -- I just searched unsuccessfully for a name for an isosceles trapezoid with three equal sides. Anybody?

*Edited on ***April 26, 2013, 10:31 am**