That's because T doesn't necessarily have to be the center of the circumcircle. However, it is theincenterof the triangle. (the incenter is the intersection of the internal angle bisectors). It's been a while since I took my transformation geometry course, but I would probably begin with an accurate drawing using a compass and straight edge.

My intuition tells me that you need to perform a dilation transformation with center T with factor -TQ/TB. then something follows from there.