I'll be referring to the colorful diagram under proof 27. There are a few things to observe.

1. The diagram consists of two parts: left and right.
2. Both on the left and on the right there are 5 colored pieces: 2 yellow, 2 red, and 1 blue. Call this square c².
3. On the left, all 5 pieces compose a square. On the right, two red pieces make a square (call it a²) while 2 yellow and the blue piece make another square (call it b²).
4. Square c² is slanted. Not to fall, it is supported by two equal right triangles with the hypotenuse exactly c.
5. On the right, those triangles are filled with colored pieces. Look carefully: again, the two triangles are equal (from the left part). On the right, one of the sides of these equal triangles is a, the other is b.
6. Since c² and a² + b² are composed of the same pieces c² = a² + b².