Three balls
Do points P and Q lie inside, on, or outside this circle?
Problem
A circle has centre O and $\angle POR = \angle QOR.$
Construct tangents at $P$ and $Q$ meeting at $T$.
Draw a circle with diameter $OT$.
Do $P$ and $Q$ lie inside, or on, or outside this circle?
Explain your answer.
Now imagine a sphere with diameter $OT$ instead. Do $P$ and $Q$ lie inside, or on, or outside this sphere? Explain your answer.
Getting Started
Here is a hint for this tough nut. You could try thinking about the angle between a chord and a tangent to a circle.
Student Solutions
As $OP$ is a radius and $PT$ is a tangent, the angle between them is a right angle.
Therefore the triangle $OPT$ is right angled. Its hypotenuse is $OT$, so $P$ lies on the circle with diameter $OT$ because a diameter always subtends a right-angle at the circumference.
In the same way, $Q$ also lies on the circle.
You can explore this by moving the red point I in the interactivity below.