Does the following provide a didactically sound approach to the Pythagorean Theorem? We first consider the hypotenuse of a right isosceles triangle and then we extend the idea to a general right triangle.
Let us consider a right isosceles triangle that has the side length $a$ and hypotenuse $c$. The Pythagorean theorem states that $2a^2 = c^2$. To see this, we divide the area of a square into two right triangles: $$a^2 = \frac{1}{2}c\cdot \frac{c}{2}+ \frac{1}{2}c\cdot \frac{c}{2}.$$
Next we consider a right triangle that has the side lengths $a,b$ and hypotenuse $c$. Here we divide the area of a rectangle into three right triangles:
$$ab = \frac{1}{2}ab + \frac{1}{2}(c-Y)X + \frac{1}{2}XY,$$
where $a/b = X/(c-Y)$ and $a/b = Y/X$ are deduced from the similar triangles. Thus, we have $ab =cX=(abc^2)/(a^2+b^2)$, i.e., the general Pythagorean theorem $a^2+b^2 = c^2$.
