Circumcircles of a trapezoid I was just wondering, what types of trapezoids have circumcircles?
I know one of them might be isoceles trapezoids, but are there any others?
 A: The necessary and sufficient condition for which a quadrilateral can be inscribed in a circle is that the opposite angles are supplementary. With reference to the image below, $$\gamma + (\pi- \beta) = \pi \Rightarrow \gamma = \beta$$ but since $AB // CD$ $$\beta = \alpha = \gamma \Rightarrow \text{the angles at the base are equals } \Rightarrow  \text{the trapezoid is isosceles}$$
So only the isosceles trapezoids can be inscribed in a circle.
A: See this picture for an Idea what the (maximum of three) possible trapezoids will look like given three points forming a circle.

Red points are candidates for extension to a trapezoid with prescribed property, the blue ones define the circle and are arbitrary.
A: If $ \ell_1, \ell_2$ are 2 parallel lines that intersect a circle at points $A, B$ and $X, Y$, then by symmetry, $AX = BY$.
Hence, the only trapezoids with circumcircles, are the isosceles trapezoids.
A: Intuitively, slide a thick transparent plastic sheet  with a parallel pair of lines drawn on it (that we normally use to project text/figures during presentations) over a fixed paper on which a sufficiently big circle is drawn. Under no circumstance is the trapezium between intersecting vertices unsymmetrical.
This is because the circle has symmetry on all its diameters and parallel lines have one line of symmetry about its common perpendicular. So in an intersection a common line of symmetry is unaffected by translation and rotation of parallel lines with respect to circle. 
