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   (x0, y0),   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, yo),   which is where the horizontal line  y = yo   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

A = (x1, y1)

and

B = (x2, y2)

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   (x0, y0)   in the line determined by   A   and   B.

top