Three Balls
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?
Age
14 to 16
Challenge level
Problem
Image
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 $OT$.
Do $P$ and $Q$ lie inside, or on, or outside this sphere?
Explain your answer. You may find an interactive hint in the second
iteractive problem:
If you can see the diagrams above, try clicking and dragging the red points.
To experiment further with this problem, download a copy of Geometer's Sketch Pad .
Getting Started
Here is a hint for this tough nut. Just think about the angle between a chord and a tangent to a circle and think about cyclic quadrilaterals.
Student Solutions
As $OP$ is a radius and $PT$ is a tangent, the angle between them is a right angle.
So the triangle $OPT$ 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.