# Are inscribable quadrilaterals always circumscribable???

• Jun 16th 2011, 07:57 AM
aldrincabrera
,.,.good day,.,i was proving "are circumscribable quadrilaterals always inscribable" and i found out that it is not,.,it only happens for some conditions,.,.but my teacher ask me the other way around,.,now i am having a hard time proving "are inscribable quadrilaterals always circumscribable" can anyone please help me with the proof???thnx
• Jun 16th 2011, 10:46 AM
Soroban
Re: Are inscribable quadrilaterals always circumscribable???
Hello, aldrincabrera!

Quote:

i was proving "Are circumscribable quadrilaterals always inscribable?"
and i found out that they are not ... it only happens for certain conditions.

But my teacher ask me the other way around. .Now i am having a hard time
proving "Are inscribable quadrilaterals always circumscribable?"

The answer to both questions is "No."
No proof is required . . . just a counterexample.

I assume a "circumscribable quadriateral" is one which can be circumscribed.
. . That is, it can be inscribed in a circle.

Code:

```              * * *         A o-----------o B         */            \*       */              \*       /                \     D o-------------------o C       *                  *       *                  *       *                *         *              *           *          *               * * *```
But no circle can be inscribed within it.

Code:

```            A o---*-*-*---------------o B             /*          *          /             *                *      /           *                *    /           /                      /         /*                  *  /         / *                  * /       /  *                  */       /                      /     /    *                *     /      *              *   /          *          */ D o---------------*-*-*---o C```
But no circle can be circumscribed around it.
. . (In this case it is, after all, a rhombus.)

In fact, the only quadrilateral which is inscribable
. . and circumscribable is a square.

• Jun 16th 2011, 12:37 PM
earboth
Re: Are inscribable quadrilaterals always circumscribable???
Quote:

Originally Posted by Soroban
...

In fact, the only quadrilateral which is inscribable
. . and circumscribable is a square.

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Well, ääh, not quite. See attachment.
• Jun 16th 2011, 03:16 PM
Soroban
Re: Are inscribable quadrilaterals always circumscribable???
Hello, earboth!

You're right, of course.

Silly me . . . I had that in my list of sketches, too.

• Jun 16th 2011, 03:32 PM
aldrincabrera
Re: Are inscribable quadrilaterals always circumscribable???
,.,as i was solving "are circumscribable quadrilaterals always inscribable" i found out this certain condition that a circumscribable quadrilateral is inscribable if its area equals the square root of the product of its four sides,.now, i am having a hard tym proving "are inscrabable...". ur sketches were really helpful but it showed me that there is a number of quadrilateral that are inscribable and circumscribable at the same time,.,.can u help me find those conditions???thnx a lot