Math CentralQuandaries & Queries


Question from Julie, a student:

Prove that tangents to a circle at the endpoints of a diameter are parallel. State what is given, what is to be proved, and your plan of proof. Then write a two-column proof.

Here is an unusual proof - but perhaps easy to 'make sense of'.

Take the center of the circle.
Do a half-turn about this center.
- the points on the circle go to points on the circle.
- the diameter goes onto the diameter.
- the two tangents trade places.

IF we assume the two tangents originally meet at a point p,
then after the half-turn, they now meet at the the half-turn image of p - call it q.
We conclude that the two lines actually meet at two points p and q.

Two lines which meet at two points (or more) are the same line! Contradiction.

We conclude the lines did not meet, and are parallel.

Walter Whiteley

About Math Central


Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.
Quandaries & Queries page Home page University of Regina PIMS