1. ## Are inscribable quadrilaterals always circumscribable???

,.,.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

2. ## Re: Are inscribable quadrilaterals always circumscribable???

Hello, aldrincabrera!

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
*                   *
*                   *

*                 *
*               *
*           *
* * *
Quadrilateral $\displaystyle ABCD$ is circumscribable.
But no circle can be inscribed within it.

Code:
            A o---*-*-*---------------o B
/*           *          /
*                *      /
*                 *     /
/                       /
/*                   *  /
/ *                   * /
/  *                   */
/                       /
/     *                 *
/       *               *
/          *           */
D o---------------*-*-*---o C
Quadrilateral $\displaystyle ABCD$ is inscribable.
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.

3. ## Re: Are inscribable quadrilaterals always circumscribable???

Originally Posted by Soroban
...

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

[/size]
Well, ääh, not quite. See attachment.

4. ## Re: Are inscribable quadrilaterals always circumscribable???

Hello, earboth!

You're right, of course.

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

5. ## 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