Any point along the first perpendicular bisector is equidistant from cities 1 and 2. Any point along the second is equidistant from cities 2 and 3. Any point along the third is equidistant from cities 3 and 1.
Since your perpendicular bisectors do not all cross at one point, there are 3 possible points which are equidistant from all 3 cities. To get them, simply select one of the three perpendicular bisectors, and calculate the distance between it and the two cities it is equidistant from. This distance will be the distance that your final point must be from the third, so simply select the point on that line which brings the distance to the third equal to that of the other two.
If there is an additional constraint (for example, the arena has to be as close as possible to all three cities), then you have to repeat this process for all three perpendicular bisectors, and choose the one whose distance from the cities is minimal.
Does this help? If not, come back and ask, and maybe if you could post the coordinates of the cities, that would help too.