the circle center will be the circumcenter of the triangle, the point formed by the intersection of the perpendicular bisectors of each side of the triangle.

on your sketch, draw in the long altitude of the triangle ... note that its length is 12.

also draw in the radii, length , from each corner of the triangle.

let = perpendicular distance from the short side (10) to the circle center.

two equations can be formed ...

or

doing the substitution ...

solve for