Perpendicular Bisector of a Chord
A chord of a circle is a segment whose endpoints are on the circle:
Theorem:
The perpendicular bisector of a chord of a circle passes through the center of the circle.
Proof:
Let the center of the circle be labeled as , the endpoints of the chord as and , the midpoint of , and the point where intersects the circle. Then all we have to prove is that .
since each is a radius of the circle. Since (is the midpoint of ) and , by SSS. Because they are corresponding parts of these congruent triangles, , so their measures are equal. But these angles are a linear pair, so their measures add to 180o: . Substituting for , we have , so . Therefore , which makes the perpendicular bisector of .