# The area A is known. What is the area B of the smallest square.

This is the accompanying figure

Given three squares as in the figure, where the largest square has area 1, and the area A is known. What is the area B of the smallest square.

Upon lookin at the answer for this excercise, I found the following reasoning as part of the solution:

Let the corner of A divide the sides of the big square in two parts x and 1-x. square B have side-length y. For the corner of B to touch the side of A, we must have y/x + y/(1-x) = 1 (the equation for a line)

I do not understand anything about the part with the line-equation, but this seems like a powerful tool i shall have use of knowing. Could someone please explain this for me?

The figure has a lot of similar right triangles. From their leg proportions, we conclude $$x:(1-x) = (x-y):y=y:(1-x-y)$$