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