# Help with a proof

• Oct 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
• Oct 27th 2008, 06:50 PM
Soroban
Hello, murphmath!

Quote:

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

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

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

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

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

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

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

. . Therefore: .\$\displaystyle AC \perp BD\$