Thread: 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

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?"
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 is circumscribable.
But no circle can be inscribed within it.

Code:

            A o---*-*-*---------------o B
             /*           *          /
            *                *      /
           *                 *     /
          /                       / 
         /*                   *  /
        / *                   * /
       /  *                   */
      /                       /
     /     *                 *
    /       *               *
   /          *           */
D o---------------*-*-*---o C

Quadrilateral 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.

Originally Posted by Soroban

...

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

[/size]

Well, ääh, not quite. See attachment.

Hello, earboth!

You're right, of course.

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

,.,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