The fact that opposite angles of the quadrilateral add to 180 degrees tells us that the quadrilateral is cyclic, ie that a circle drawn through any three vertices will also pass through the fourth.

The fact that the four vertices can lie on a circle lets us use a fact about lines crossing in circles: the product of two opposing distances from a particular point within a circle to the edge of the circle will always be the same, regardless of the angle of the line through the point.

In the following diagram for instance, ab = cd, as a general rule.

We can use this fact if we take the crossing points as the centre of the circle, and the four distances as (12+r), (31+r), (27+r) and (10+r).

(12+r)(27+r) = (10+r)(31+r)

324 + 39r + r^2 = 310 + 41r + r^2

14 = 2r

r=7

So the radius of the circle is 7.