It's been a day or two? I did not keep track. Found this one in an online conversation I had with a geometer friend of mine; it might be or based on an existing olympiad problem though. I never keep track anyway.
\(\triangle ABC\) has a right angle at \(A\), \(D\in BC\) and \(AD\perp BC\). \(X\) is an arbitrary point of \(AD\). \(E,F\in CX,BX\) respectively such that \(AB=BE,AC=CF\).
Suppose \(BE\cap CF=G\). Prove that \(FG=GE\).
Extra Credit: Let \(XG\cap BC=Q\). Prove that \(E,F,D,Q\) are concyclic.