Theorem 3.9: Let A and B be two points on a circle. The foot of the center of the circle in the line determined by A and B is the midpoint of the line segment between A and B.

Proof: Let the center of the circle, be (x_o, y_o), and let r be the radius of the circle.

If A and B are vertical then A = (a, y₁) and B = (a, y₂) for some real number a. Then, in this case, the x-coordinate of the midpoint of the line segment between A and B will be a. By Theorem 3.2 we can take, after possibly relabeling,

and

So

and the midpoint of the line segment between A and B will be (a, y_o), which is where the horizontal line y = y_o meets the vertical line x = a which will be the foot of the center in the line determined by A and B.

If A and B are not vertical, then their line has an equation of the form y = mx + b by Theorem 1.2. Let

and

By Theorem 3.3, we could take, after possibly relabeling,

and

So

Then

and

so

This gives us the coordinates of the midpoint of the line segment between A and B to be

which, by Theorem 1.10 are the coordinates of the foot of (x_o, y_o) in the line determined by A and B.

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