# Math Help - Diagonals of Cyclic Quad

1. ## Diagonals of Cyclic Quad

Prove that if the diagonals of a cyclic quadrilateral are equal then so is a pair of opposite sides.

2. Hello, cedricc!

Prove that if the diagonals of a cyclic quadrilateral are equal ,
then so is a pair of opposite sides.
Code:
              * * *
*           *
A o---------------o B
*  *            *|*
*         * |
*       *      *  | *
*         *   *   | *
*           **    | *
* *   |
*         *    * |*
*       *       o C
*    *      *
o * *
D

Draw $AD$ and $CD.$
We have cyclic quadrilateral $ABCD.$

$\begin{array}{ccccc}1. & \overline{AC} \:=\: \overline{BD} & & 1. & \text{Given} \\ \\

2. & \text{arc}(ADC) \:=\: \text{arc}(BCD) & & 2. & \text{Equal chords = equal arcs} \\ \\

3. & \text{arc}(CD) \:=\:\text{arc}(CD) && 3. & \text{Identity axiom} \\ \\

4. & \text{arc}(AD) \:=\:\text{arc}(BC) && 4. & \text{Subtraction postulate} \end{array}$

$\begin{array}{ccccc}5. & \qquad \quad\overline{AD} \:=\:\overline{BC} && \qquad\quad 5. & \text{Equal arcs = equal chords.}
\end{array}$