Before exploring adaptations of Euclid’s strategy to figures other than squares, let's recall his proof. We repeat Figure 1 for convenience.

**Figure 1** (repeated from the preceding page)**:** The Theorem of Pythagoras from Euclid's *Elements* with the "Pythagorean cut," line segment *LK,* shown in red.

Euclid showed how to partition the largest square in Figure 1, above, into two rectangles so that each rectangle would have the same area as one of the two smaller squares. The proof is essentially this. Drop a perpendicular from *B* to *JH* hitting *JH* at *K* and *AC* at *L *and draw construction lines* DC *and* BJ *(see Figure 1). Then triangle *DAC* is congruent to triangle *BAJ* by *SAS* and so the two triangles have the same area. Moreover, the area of triangle *DAC* is half the area of square *BADE* since they share the same base *DA* and have the same altitude *AB*. Likewise, the area of triangle *BAJ* is half the area of rectangle *AJKL* since they share the same base *AJ* and have the same altitude *AL*. Thus, the area of square *BADE* must be the same as the area of rectangle *AJKL*. Similarly, the area of square *CGFB* equals the area of rectangle *LKHC* based on congruent triangles *ACG* and *BCH*. So, segment *BK* partitions, or cuts, the largest square into two rectangles, each of which has area equal to the area of one of the other two squares. We say that segment *LK* is the “Pythagorean cut” for square *AJHC*. The Pythagorean cut, separating the largest square into two figures each of which has area equal to one of the two smaller squares, is shown in red in Figure 1.

Underlying Euclid's proof is his simple but profound observation in I.35 [Heath, p. 326]:

*Parallelograms which are on the same base and in the same parallels are equal to one another.*

From here, it is a fairly easy path to these two key observations:

*Triangles which are on the same base and in the same parallels are equal to one another. *(I.37 [Heath, p. 332])

*If a parallelogram have the same base with a triangle and be in the same parallels, the parallelogram is double of the triangle*. (I.41 [Heath, p. 338])

For example, in Figure 1, using lines through *EC* and *DA* as the parallels, triangle *DAC* and square *DABE* certainly fit the criteria in I.41.