I am sorry first, my mother language is not English so I have a problem to explain ‘what I want to know’. But I will try my best.
The Pythagorean rule said that “C^2=ab”. And I understood if a=b, then it become a=b=c. But if a and b is not the same length, why C^2 still is ab?? I want to understand the prove. Thank you for reading.
All of the triangles are similar so the ratios of their sides are the same. This means we can multiply both sides by the denominator product and get
$$\dfrac{a}{c}=\dfrac{c}{b}\implies a\cdot b=c\cdot c$$
The smallest Pythgorean triple with an integer for diagonal $\,c\,$ is shown here. Using extended manipulation of the Pythagorean Theorem, we can derive $\quad c=\dfrac{AB}{C}\quad$ which can also be verified by inspection of this example.