# Proof 4.11

Prove that the perpendicular bisectors of a quadrilateral are concurrent if and only if the quadrilateral is cyclic.

[ ] Let ABCD be a quadrilateral with concurrent perpendicular bisectors (Proposition 10). Let the point of concurrency be I. Let the midpoint of be E, so . By construction, because they are perpendicular. So by SAS (Proposition 4), . This means that . Similarly, all of the vertices of the quadrilateral are equal length from point I. Therefore by the definition of a circle, we know that the quadrilateral ABCD is cyclic based on the definition of a cyclic quadrilateral.

In conclusion based on the two arguments above, we can conclude that the perpendicular bisectors of a quadrilateral are concurrent if and only if the quadrilateral is cyclic.

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