Let ABCD be a quadrilateral such that AB is perpendicular to BC, AD is perpendicular to BD and $$AB = BC$$, $$BD = a$$, $$AD = c$$, $$CD = x$$. Find $$x$$ in terms of $$a$$ and $$c$$.

PLACEHOLDER ANSWER - the answer is an expression. Take $$B(0, 0)$$ and $$D(a, 0)$$; since $$AD \perp BD$$, $$A = (a, c)$$. C is A rotated through $$90^\circ$$ about B, and for ABCD to be a proper (non-crossing) quadrilateral C lies on the other side of BD, so $$C = (c, -a)$$. Then $$x^2 = (a - c)^2 + a^2$$, i.e. $$x = \sqrt{2a^2 - 2ac + c^2}$$.