The Pythagorean theorem says that for any right-angled triangle, the lengths
(1)for sides $a$, $b$, and diagonal $c$.
Because
The area of a square with side $a$ is equal to $a^2$. The area of a square with side $b$ is equal to $b^2$. The area of a square with side $c$ is equal to $c^2$.
The area of the smaller squares ($a^2 + b^2$) equals the area of the larger square ($c^2$).
Because
You can divide the largest square ($c^2$) into two rectangles. One rectangle has the same area as one smaller square ($a^2$). The other rectangle has the same area as the other smaller square ($b^2$).
Because
You can start with the three sides of the triangle to form (external) squares that include sides $a$, $b$, and $c$.
Divide the largest square into two rectangles using a line that also separates the two smaller squares from each other.
Each rectangle has the same area as the nearest square.
Because
For a given side of the diagram, the square has the same area as the rectangle.
Because
The area of half the rectangle is the same as the area of half the square.
Because
A triangle that's half the square has the same area as a triangle that's half the rectangle.
Because
The square's triangle has the same area as two congruent triangles.
The rectangle's triangle has the same area as two congruent triangles.
Because
The square's triangle has the same area as a triangle with one side the same as the smaller square's, and another side the full side of the large square.
The rectangle's triangle has the same area as a triangle with one side the same as the smaller square's, and another side the full side of the large square.
These two new triangles are congruent.
Because
The area of a triangle is equal to the base times height divided by two. Each of the two new triangles has the same base and height as the original triangle.
Two triangles are congruent if they have two congruent sides and the angle between them is identical.
Each of the two new triangles has two sides that are the same length as each other's, and the same angle between those two sides.
Because
The angle between the two sides is a right angle plus the same angle from our original triangle.
Notes
I'm playing pretty fast and loose with notation for the sake of brevity. Technically side $a$ and measurement of the length of side $a$ should be distinguished (say, by calling the latter ${\mathrm m}(a)$).
Also, I may have gotten the italics wrong here (a label isn't a variable), but I'll deal with that later. Some more diagrams might be useful too.
More Info
Check out http://www.cut-the-knot.org/pythagoras/.
Previous Version
Here is an earlier rewriting that I attempted. I believe it is accurate. I just find it wordy.
The Theorem
The Pythagorean theorem says that for a right-angled triangle,
(2)where $a$, $b$, and $c$ are the sides of a right-angled triangle—and $c$ is the diagonal.
A Reverse Belief-based Proof
Consider a right-angled triangle. Two of the sides form a right angle. Square the length of one side and add it to the square of the other side. The sum will equal the square of the third side, the diagonal.
If you believe…
After forming a square on each of the triangle's three sides, the largest square's area will equal the sum of the two other squares' areas.
If you believe…
After dividing the largest square into two rectangles, one of the smaller squares will have the same area as one rectangle, and the other of the smaller squares will have the same area as the other rectangle.
If you believe…
Half of one rectangle equals half of one small square, and half of the other rectangle equals half of the other small square.
If you believe…
There is a straight line that will divide the large square into two rectangles and separate the two small squares from each other. Consider either side of the line. The area of a triangle that's half of the large square will equal the area of a triangle that's half of the adjacent small square.
If you believe…
A triangle that's half the rectangle has the same area as another triangle that shares one side with the large square and another side with the nearest small square.
That second triangle has the same area as another triangle that also shares one side with the large square and one side with the previously mentioned small square.
That third triangle has the same area as the triangle that's half the small square.
Therefore the first triangle has the same area as the fourth.
If you believe…
Two triangles with the same base and height have the same area.
If you believe…
The area of a triangle is equal to its base times height divided by two.
Advisory
Please note that I am not a mathematician and so the presentation of proofs that I make may be deeply flawed. I'm using this writing process to figure out what I'm reading. Please consult more authoritative sources as well.
Feel free to contact me by leaving a comment or sending me a private message.
