> it follows that...

"Now just draw the rest of the owl."

Does it not feel like you skipped something here? The areas add up and area scales quadratically, therefore... Pythagorean Theorem? It definitely is not clear how this follows, even after the questionable assumption that it's obvious area scales quadratically.

He didn't skip anything but he left the details to the reader:

Let C be the area of the big triangle, A and B be the areas of the two small triangles. By construction we know that C = A + B. Moreover, a, b, c are the hypotenuses of the triangles A, B and C.

The area scaling quadratically with the similarity ratio means that

A = (a/c)² C, and B = (b/c)² C.

Now, plug this into A + B = C, cancel C, rearrange.