**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*_{0}, *y*_{0}), and let *r *be the radius of the
circle.

If *A *and *B *are vertical then *A* = (*a*,
*y*_{1}) and *B* = (*a*, *y*_{2}) 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 the
line between them 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*_{0},
*y*_{0}) in the line determined by *A* and *B*.