• Jun 9th 2010, 06:09 PM
cedricc
Prove that if the diagonals of a cyclic quadrilateral are equal then so is a pair of opposite sides.
• Jun 10th 2010, 08:20 AM
Soroban
Hello, cedricc!

Quote:

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}$

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