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Transformation of two squares in a square

We use the squares in the figure:

Draw segment BE and perpendicular straight lines r and s to the EB passing,
respectively, to points I and H.

Let H be the point of r on segment
FG.

Draw the perpendicular straight line to the EH passing through H and
let I be the point of intersection of this straight line with s.

Draw IJ perpendicular to the BG.

The colors indicate the pieces to transform the two squares given, ABCD
and ECGF, into the biggest square BEHI.

As before, intuitively we can see that the parts
cut in the two lesser squares are equal in the bigger square. Formally
we must show the congruences between the pairs of triangles LAB and MJI,
HGM and EDL, and FEH and JBC.