Given AB and CD are two chords of a circle C(O,r) and let the perpendicular bisectors O'E and O'F of AB and CD respectively meet at O'.
To prove O' coincide with O.
Construction join OE and OF.
proof :-E is the midpoint of Chord AB
⇒ OE ⊥ AB
⇒ OE is the perpendicular bisector of AB
⇒OE as well as O'E is the perpendicular bisector of AB
⇒O'E lies along OE.
Similarly, F is the midpoint of Chord CD
⇒ OF ⊥ CD
⇒ OF is the perpendicular bisector of CD
⇒OF as well as O'F is the perpendicular bisector of CD
⇒O'F lies along OF.
Thus, O'E lies along OE and O'F lies along OF
⇒The point of intersection of O'E and O'F coincides with the point of intersection of OE and OF
⇒ O' coincides with O
Hence,the perpendicular bisectors of AB and CD intersect at O.
