Point Q is the circumcenter of triangle UWV.
This would have been a much easier problem if you had bothered to actually include a diagram. But given the description of the line segment lengths, you have a triangle with vertexes UWV and the points T, S, and R are bisectors of their respective sides. Point Q is the circumcenter of triangle UWV. One easy way to find the circumcenter of a triangle is to construct the perpendicular bisector of the triangle's sides and the point of intersection of those bisectors is the circumcenter.