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

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?"

Can anyone please help me with the proof?

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*.

1 Attachment(s)

Re: Are inscribable quadrilaterals always circumscribable???

Quote:

Originally Posted by

**Soroban** ...

In fact, the **only quadrilateral** which is inscribable

. . *and* circumscribable is a *square*.

[/size]

Well, ääh, not quite. See attachment.

Re: Are inscribable quadrilaterals always circumscribable???

Hello, earboth!

You're right, of course.

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

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