Intersection of a Line and a Circle: The Common Points Theorem

A line and a circle can intersect at no more than two points.

Put simply, at most two common points can exist where a line intersects a circle.

A line that intersects the circle at two points is termed a secant line.

A line that intersects the circle at just one point is known as a tangent line.

 

If a line does not intersect the circle at all, it is referred to as an external line.

Demonstrating the Theorem

Let's consider a proof by contradiction to demonstrate this theorem.

Assume line r and the circle intersect at three or more points: A, B, and C.

These points A, B, and C all lie on the same line, r, and form aligned segments AB and BC.

Being aligned, the segments AB and BC have their axes perpendicular to line r and parallel to each other.

The axes MM' and NN' of the segments AB and BC are parallel, and thus do not intersect each other.

$$ \overline{MM'} \cap \overline{NN'} = \emptyset $$

However, segments AB and BC are also chords of the circle, implying that points A, B, and C are not only points on the line but also intersect with the circle.

Given that the axis of any chord passes through the circle's center, we can conclude that the axes of chords AB and BC intersect at point O.

Therefore, axes MM' and NN' intersect at point O, contradicting the premise that they are parallel.

$$ \overline{MM'} \cap \overline{NN'} = O $$

This contradiction proves that a line and a circle cannot have three common points.

Note: This reasoning also applies if we assume there are four or more points of intersection between the line and the circle.

If the initial hypothesis is false, the converse must be true.

Thus, a line and a circle can intersect at most at two points.

So it goes.