Dudeney

Dudeney's dissection of an equilateral triangle

Dudeney's dissection of an equilateral triangle into a square.
Triangle t = (ABC) is equilateral. D, E are the middles of its sides. F and G the projections of E, D on AB. The three quadrangles (1), (2), (3) created and the triangle (4) are linked at D, E and F.

In the figure above, the 4 points (1), (2), (3), (4) and X are control-points, that enable the movement of the corresponding polygons. You can catch them and trasnsform the picture to an equilateral triangle or a square.
Dudeney discovered this in 1902. More on this you can read in the book:
[Howard Eves, A survey of Geometry, Allyn & Bacon, Boston 1963, p. 260]
For another example, that moves linked polygons look at the file: Clifford_Cayley.html .

http://users.math.uoc.gr/~pamfilos/