Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

how can i determine wether a circle can be circumscribed about a quadrilateral?

share|cite|improve this question
Look at opposite angles. A convex quadrilateral is cyclic if and only if its opposite angles add up to $\pi$ ($= 180^\circ$). – t.b. Mar 14 '11 at 17:49
See – lhf Mar 14 '11 at 18:23

If you're given a convex quadrilateral, a circle can be circumscribed about it if and only the quadrilateral is cyclic. A nice fact about cyclic quadrilaterals is that their opposite angles are supplementary.

Proposition III.22 of Euclid's Elements gives a proof that the opposite angles of cyclic quadrilaterals are equal to two right angles. The converse is also true, that if the opposite angles of a quadrilateral are supplementary, then the quadrilateral is cyclic.

Another way to identify if a quadrilateral is cyclic is given in Hartshorne's book on classical geometry. A nice proof can be find in Hartshorne's Euclid: Geometry and Beyond, which I will include here.

enter image description here enter image description here

If you can determine either of these facts hold about your quadrilateral, then you know there exists a possible circle circumscribed about it.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.