Using Pythagoras Theorem twice we get $a^2 + c^2 = k^2$ and also $(a + b + c)^2 = 1 + k^2$
By some routine algebra we get $a^2 = k^2/(1 + k^2)$ and the area of the inner square is $b^2 = (1 - k)^2/(1 + k^2).$