I have this diagram.

I know that PCQ = 45 Degrees and Angle C = 90 Degrees.

I want to prove that AB^2 + BQ^2 = PQ^2.

ugh.... I don't know what to do next.

Help would be greatly appreciated fast :D

You cannot prove it because it is not true. I assume you have written the question incorrectly.

You want to prove that AB^2 + BQ^2 = PQ^2.

But AB is bigger than PQ so the statement to be proven is not true

Sorry about that!

I meant to say AP^2 + BQ^2 = PQ^2

AC = BC = 5 | ∠ACB = 90º | ∠PCQ = 45º | ∠ACP = ∠BCQ = 22.5º

Let M be a midpoint of AB

AB = sqrt (AC^{2} + BC^{2})

PQ = 2 * [tan(PCM) * CM] ( CM = AM = BM )

AP = BQ = 1/2 (AB - PQ)