Circle Geometry and Tangents

A tangent touches the circle at one point.

• Join both ends of a diameter to any other point on the circle. The angle there is 90 degrees: the angle subtended by a diameter.
• The perpendicular from the centre to a chord bisects it (splits it into two equal lengths).
• A tangent is perpendicular to the radius at the point of contact.

The second fact means that the perpendicular bisector of a chord passes through the centre. Two non-parallel chord bisectors therefore locate an unknown circle centre.

The normal is perpendicular to the tangent at the contact point. For a circle, the normal is the radius line, so it passes through the centre.

1. Find the centre and the point of contact.
2. Find the gradient of the radius. This is the normal gradient.
3. Take its negative reciprocal to get the tangent gradient.
4. Use the point of contact in point-gradient form.

If the radius at the contact point is horizontal, the tangent is vertical. If the radius is vertical, the tangent is horizontal. Use this geometry directly because one gradient is undefined.

A quick check is to substitute the contact point into both the circle and the tangent equation. It must lie on both, and the tangent and radius directions must form a right angle.