Chris, age 15, was able to solve the first part:

As $\mathrm{OP}$ is a radius and $\mathrm{PT}$ is a tangent, the angle between them is a right angle. So the triangle $\mathrm{OPT}$ is right angled, and the hypotenuse is the diameter of the circle, so $P$ lies on the circle. In the same way, $Q$ lies on the circle.