The Proof

Let A, B, and C be the sides of a triangle. We want to show that A² + B² = C².

Assume the Triangle

We start by assuming that the triangle is, in fact, a triangle.

Let's say A = BC = CD = DE.

Then A = DE, B = EF, and C = FD.

Draw a diagram to represent the situation:

We have two similar triangles: ABC and DEF.

We know that similar triangles have proportional sides.

We can set up the following proportions: A = DE, B = EF, and C = FD.

We can cut the triangle ABC into two smaller triangles by drawing a line from DE to C.

This line divides the triangle into two smaller triangles: ABE and DEA.

We can write A² + B² + C² = AB² + BC² + CA².

We know that AB² = DE² + EF² = A² and BC² = FD² + DE² = C².

We also know that CA² = FD² + DE² = C².

This means that A² + B² + C² = AB² + BC² + CA².

And thus, we have proved that A² + B² = C².

Q.E.D. (Quod Erat Demonstratum, or "Thus it was proven".)