Proposition 47

You can drag triangles and parallelograms around in this figure to understand Euclid's proof of the Pythagorean theorem, Proposition 47 of Book I of his Elements. The triangle ABC is a right triangle. Euclid continues the triangle altitude from C, dividing the square on the hypotenuse AB into rectangles (colored blue and gold) having areas equal to the squares on the corresponding sides: Thus, the area of the square on the hypotenuse is the sum of the areas of the squares on the other two sides. Each individual sliding or turning move you make clearly preserves area, but there are just enough steps that the conclusion is not at all obvious.

A B C
  1. The proof must work for any right triangle, so you may first drag point C to select the angles you want.
  2. Click on any of the four colored regions to begin. Your region will darken and its corresponding region, which we want to prove equal, will bleach to white.
  3. Slide the central white triangle outward over your darkened region, converting it into a parallelogram of equal area.
  4. Rotate this parallelogram ninety degrees. Because its sides initially coincide with the edges of the squares on the hypotenuse and one side, the ninety degree rotation leaves the sides coincident with the adjacent edges of their squares.
  5. Slide the white triangle inward over the parallelogram. You have transformed your initial darkened region into its matching region without changing its area, thus proving the matching areas equal.
  6. Click on the darkened region to return to the initial state.
A B C H

Euclid's argument is elegant because he shows us how to partition the square on the hypotenuse into parts which separately equal the squares on the sides. Furthermore, the altitude from point H on the the hypotenuse AB to the right angle C connects algebra and geometry in other important ways. For example, the height HC is the geometric mean between HA and HB. Even more important in wave physics is the reciprocal Pythagorean theorem: The sum of the squares of the reciprocals of the sides AC and BC equals the square of the reciprocal of the height HC.

Elegance and foreshadowing aside, Euclid's proof does go beyond the bare statement of the proposition. We don't really need to partition the square on the hypotenuse. Here is a well-known proof which sheds that baggage, not distinguishing between the areas of the squares on the sides at all.

  1. Again, you may first drag the intersection point where the two blue squares touch to select the shape of the four right triangles.
  2. In the initial configuration, you may also drag the upper left white rectangle downward to rotate it by ninety degrees, checking that the diagonals of the white rectangles are ninety degrees apart.
  3. Split the lower right white rectangle, sliding one of its triangles left and the other up as far as possible.
  4. Split the upper left white rectangle by sliding its central triangle down and right as far as possible, creating a tilted blue square in the center.

The shape of the blue area has changed from two squares whose sides are the sides of the right triangles into a single square whose side is the hypotenuse of the right triangles, proving the proposition. The adjacent sides of the tilted square in the final configuration are perpendicular because the two white rectangles in the initial configuration, including their diagonals, are identical after a ninety degree rotation.

The fact that translating (sliding) or rotating any shape does not change its area is really a fundamental principle of physics. The axioms of geometry come from our experience of the world around us. As Newton puts it in the preface to Principia:

To describe right lines and circles are problems, but not geometrical problems. The solution of these problems is required from mechanics, and by geometry the use of them, when so solved is shown; and it is the glory of geometry that from those few principles, brought from without, it is able to produce so many things. Therefore geometry is founded in mechanical practice, and is nothing but that part of universal mechanics which accurately proposes and demonstrates the art of measuring.