# Help with a proof

• October 27th 2008, 06:06 PM
murphmath
Help with a proof
Could I please get some help with the following?
Given a quadrilateral with points labeled sequentially A B C D with a line segment AC and a line sement BD in the interior noted at intersection by point E with the following givens: segment BC = segment CD and segment AB = segment AD

Prove that AC is perpendicular to BD I'm stuck!
thanks SO much for any help or hints
• October 27th 2008, 06:50 PM
Soroban
Hello, murphmath!

Quote:

Given a quadrilateral $ABCD$ with diagonals $AC$ and $BD$ intersecting at point $E$,
with the following givens: $BC = CD$ and $AB = AD.$

Prove that $AC$ is perpendicular to $BD.$

We have a kite-shaped figure . . .
Code:

```            A             *           * | *         *  |  *     D * - - + - - * B       *    |E  *         *  |  *         *  |  *           * | *           *|*             *             C```

We are given: . $AB = AD,\;BC = CD$

Point $A$ is equidistant from $B$ and $D.$
Point $C$ is equidistant from $B$ and $D.$

Hence, $AC$ is the perpendicular bisector of $BD.$

. . Therefore: . $AC \perp BD$