It is easy to divide a square into four congruent isosceles triangles, just draw both diagonals. But can you divide a square into four incongruent isosceles triangles?

From the top left going clockwise, label the corners of the square A,B,C,D.

Draw a diagonal from A to C. The first isosceles triangle is in the upper right-- triangle ABC.

From A, lay off along the diagonal a length equal to one side of the square (call it length 1, for a unit square), and label the point E. The second isosceles triangle is AED.

The remaining length of the diagonal, EC, is sqrt(2) - 1.  Construct a circle of that length centered on E and label the intersection with side CD, F. Triangle ECF is the third isosceles triangle.

And according to GSP, EF = FD, making EFD isosceles also.

Posted by Charlie on 2007-03-23 14:43:59