Let
A, r1 = center, radius of first circle
B, r2 = center, radius of second circle
be the given input data and
C, r3 = center, radius of third circle
be the largest circle that fits into the intersection of the first two circles.
Denote by
D = intersection point of first with third circle
E = intersection point of second with third circle
D and E are points on the line connecting the centers A and B. D has distance r1 from A and E has distance r2 from B. Therefore
D = A + r1 * (B - A)/dist(A, B)
E = B - r2 * (B - A)/dist(A, B)
from which follows
C = (D + E)/2 = (A + B + (r1 - r2)*(B - A)/dist(A, B)) / 2
r3 = dist(D, E)/2 = (r1 + r2 - dist(A, B)) / 2
If r3 < 0
then the circles do not intersect at all.
(The above calculation assumes that none of the circles lies completely within the other circle.)