An Arc Angle Angle Theorem

Angles, Circle
The student is free to move points P, Q, and R around, and visually verify that angle PRQ appears to be half of angle PTQ, in most configurations. In particular, the angle PRQ, surprisingly, stays constant as we move the point R, so long as R is outside of the angle PTQ! Inspired by B Presley.
Define conditions such that the measure of angle PRQ is half that of angle PTQ. Prove that PQR = 1/2 PTQ does indeed hold in those conditions. (Feel free to make helpful constructions) In what configurations does this not hold? Why? What property holds in those configurations? Prove?