# Paralellogram

• Aug 20th 2008, 10:35 PM
devi
Paralellogram
Any cyclic paralellogram having unequal adjacent sides is necessarily a
(a)Square
(b)Rectangle
(c)Rhombus
(d)Trapezium
Why?
• Aug 21st 2008, 01:54 AM
Simplicity
If a parallelogram is cyclic, then it must be a rectangle. The perpendicular bisectors of the sides of a parallelogram are concurrent only when it is a rectangle. For a parallelogram to be cyclic, its perpendicular bisectors must be concurrent at the center of the circumcircle; therefore, a cyclic parallelogram must be a rectangle.
• Aug 21st 2008, 05:23 AM
Soroban
Hello, devi!

Quote:

Any cyclic paralellogram having unequal adjacent sides is necessarily a:
. . (a) square . (b) rectangle . (c) rhombus . (d) trapezium

This is my approach to this problem.
Perhaps you can hammer it into a formal proof.

Opposite sides of a parallelogram are parallel.
They are formed by a pair of parallel chords.
Code:

```              * * *           *          *         *              *     A o o o o o o o o o o B                         *                  *       *        *        *     D o o o o o o o o o o o C                         *                *         *              *           *          *               * * *```

Opposite sides of a parallelogram are equal.
They are formed by two chords equidistant from the center.
Code:

```              * * *           *          *       A o o o o o o o o o B       *        :        *                 :       *        :        * C       *        *        o       *          \      o *                   \  o       *            o    *         *        o    *           *    o    *               o * *             D```

Since opposite sides of a parallelogram are parallel and equal,
the chords are parallel and equidistant from the center.
Code:

```              * * *           *          *       A o o o o o o o o o B       *        :        *                 :       *        :        *       *        *        *       *        :        *                 :       *        :        *       D o o o o o o o o o C           *          *               * * *```