Let's consider a right triangle and draw the altitude to the hypotenuse. We will label all the segments formed.
By applying the Geometric Mean (Leg) Theorem, we can write the following two equations.
c/b = b/x
and
c/a = a/y
From the first equation we have that b^2 = c x, and from the second equation we get a^2 = c y. Next, let's add these two equations.
b^2 &= c x
+ a^2 &= c y
a^2 + b^2 &= c( x + y)
From the diagram and by the Segment Addition Postulate, we have that x + y = c. Let's substitute this into the latter equation.
a^2 + b^2 = c c
⇓
c^2 = a^2 + b^2
The final expression is what the Pythagorean Theorem states.
