[b]Forwards implication: [/b]If the perpendicular bisectors of a quadrilateral are concurrent, then the quadrilateral is cyclic. [br][b]Proof of forwards implication (by direct proof): [/b]Consider the quadrilateral ABCD. Assume that the perpendicular bisectors of ABCD are concurrent. That is, the perpendicular bisectors of ABCD all intersect at some point, P. [br]Consider the side AB of the quadrilateral. Because any point on a perpendicular bisector of a line segment must be an equal distance from each endpoint (proved in class), we can assume that [math]AP\cong BP[/math]. [br]Similarly, when considering side BC, we can assume that [math]BP\cong CP[/math]. [br]When considering side CD, we can assume that [math]CP\cong DP[/math]. [br]Finally, when considering side DA, we can assume that [math]DP\cong AP[/math]. [br]That is, [math]AP\cong BP\cong CP\cong DP[/math]. Thus, the points, A, B, C, D lie on a circle centered at P. Therefore, the quadrilateral ABCD is cyclic. [br][br][b]Backwards implication: [/b]If a quadrilateral is cyclic, then the perpendicular bisectors of the quadrilateral are concurrent. [br][b]Proof of backwards implication (by direct proof): [/b]Consider the cyclic quadrilateral ABCD.By definition, we know that the points A, B, C, D must lie on a circle such that A, B, C, D are all equal lengths from the center of the circle, P. That is, [math]AP\cong BP\cong CP\cong DP[/math]. [br]Given that a point lies on a perpendicular bisector of a segment if it is an equal distance from both endpoints (proved in class) and [math]AP\cong BP[/math], we can conclude that P is on the perpendicular bisector of side AB. [br]Similarly, because [math]BP\cong CP[/math], we can conclude that P is on the perpendicular bisector of side BC. Because [math]CP\cong DP[/math], we can conclude that P is on the perpendicular bisector of side CD. Finally, because [math]DP\cong AP[/math], we can conclude that P is on the perpendicular bisector of side DA. [br]Because P lies on every perpendicular bisector of quadrilateral ABCD, we know that the perpendicular bisectors are concurrent at point P.