klein view page

tangents to a circle

The Circle
For any circle C, each family of parallel lines contains exactly two tangents
to C, and these occur at diametrically opposite points of C.

Proof
Suppose that L is a line. Let AB be the diameter of C perpendicular to L.
Let M be a line parallel to L.

If M does not meet AB, then it does not meet the circle.
If M cuts AB between A and B, then it cuts the circle twice -
once on each side of AB.
If M cuts AB at A or B, then it meets the circle only at this point.
To verify this, suppose that M meets C again, at P say. Then it
will meet C at a third point - the reflection of P in C. But a line
cuts a circle at most twice, so M meets C exactly once. It is the
tangent to C at this point.

special conics page

The Circle For any circle C, each family of parallel lines contains exactly two tangents to C, and these occur at diametrically opposite points of C.
Proof Suppose that L is a line. Let AB be the diameter of C perpendicular to L. Let M be a line parallel to L. If M does not meet AB, then it does not meet the circle. If M cuts AB between A and B, then it cuts the circle twice - once on each side of AB. If M cuts AB at A or B, then it meets the circle only at this point. To verify this, suppose that M meets C again, at P say. Then it will meet C at a third point - the reflection of P in C. But a line cuts a circle at most twice, so M meets C exactly once. It is the tangent to C at this point.