Question 563454: a) Prove the diagonals of a cyclic quadrilateral bisect each other
b) prove if the diagonals of a quadrilateral bisect each other then the quadrilateral is cyclic.
I know the opposite sides of the quadrilateral are equal and the SAS theorem proves the triangles made by the diagonals are equal I just dont know how to write the proofs
Answer by richard1234(7193)   (Show Source):
You can put this solution on YOUR website! Statements a) and b) cannot be true, since it is possible to find counterexamples for each one. Note that a cyclic quadrilateral is a quadrilateral whose four vertices all lie on a circle.
For statement a), a "kite" shape formed by taking a right triangle and reflecting it about the hypotenuse is a counterexample, since such a quadrilateral is cyclic and not every kite shape will have diagonals that bisect each other.
For statement b), a parallelogram (other than a rectangle) is a counterexample, because its diagonals bisect each other, but parallelograms (other than rectangles) *cannot* be cyclic (since opposite angles must add to 180).
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