Euclid Book I Proposition 47. The Pythagorean Theorem.
- James J Madden
This is an illustration of a variant of Euclid's proof of the Pythagorean Theorem. The image can be adjusted by dragging the point where the two small squares meet. The orange square (partly covered by the yellow parallelogram) and the orange rectangle (partly covered by the green parallelogram) have equal area. We can see this by examining the two parallelograms. They are congruent to one another, since one is obtained by rotating the other by 90 degrees about the vertex that they have in common. The green parallelogram the overlaps the orange rectangle has the same area as that rectangle by Euclid, Book I, Proposition 35: Parallelograms on the same base and between the same parallels have equal area. The yellow parallelogram overlaps the orange square has the same area as the square, also by Proposition 35.
If we divide the big square into rectangles in given ratio (with respect to area), then the small squares will have the same ratio (with respect to area). Position the moveable point so that the purple square has (a) twice the area of the orange one, (b) three times the area of the orange one, (c) 5 times the area of the orange one (d) 11 times the area of the orange one.