Commentary on Proposition I.46¶
Back to Proposition I.46.
Proclus (pp. 424—5) proves that, if squares are described on equal straight lines, the squares are equal; and, conversely, that, if two squares are equal, the straight lines are equal on which they are described. The first proposition is immediately obvious if we divide the squares into two triangles by drawing a diagonal in each. The converse is proved as follows.
Place the two equal squares AF, CG so that AB, BC are in a straight line. Then, since the angles are right, FB, BG will also be in a straight line. Join AF, FC, CG, GA.
Now, since the squares are equal, the triangles ABF, CBG are equal.
Add to each the triangle FBC; therefore the triangles AFC, GFC are equal, and hence they must be in the same parallels.
Therefore AG, CF are parallel.
Also, since each of the alternate angles AFG, FGC is half a right angle, AF, CG are parallel.
Hence AFCG is a parallelogram; and AF, CG are equal.
Thus the triangles ABF, CBG have two angles and one side respectively equal; therefore AB is equal to BC, and BF to BG.