Two overlapping circles with tangents drawn at their intersection points intersecting at each others' centres. 
So I'm stumped by what should be a rather simple problem. There are two circles whose tangents intersect at each others' centres. The tangents are at right angle. If I know the distance between the centres, there should be simple geometry to solve the radii of the circles.
I know I could do like an equation group of this; call radius one $a$, radius two $b$, write down Pythagoran theorem for the triangle in the middle, then maybe like trigonometric functions for the halves of the two central angles of the circles, but I can't believe it should be this complex (that would be what, a four equation group?). There's something about the symmetry of these circles I'm missing that ought to make this simpler.
I know I have:
Let the radius of circle $A$ be $a$, and circle $B$ be $b$. Further, let the line drawn at their intersection be $c$, and the line from $c$ to centre of circle $B$ be $d$.
$$a^2 + b^2 = 10^2 \\a^2 + (10-d)^2 = \left( \frac c 2 \right) ^2 \\
b^2 + d^2 = \left( \frac c 2 \right) ^2 $$
If I add to that trigonometric functions and the sum of the angles, I believe I could solve them but it just feels way too complex and certainly not the natural solution.
An aside: does anyone know how these circles (ones whose tangents drawn at the intersection points of the circles are the centres of one another) are called? I know there's a term for it but I can't find it for the life of me.

Let the centers of the circles be $O_1$ and $O_2$. Draw a circle with the diameter $O_1O_2$. Take an arbitrary point $C$ on the circle. Draw through $C$ two circles centered at $O_1$ and $O_2$. The circles will be orthogonal (i.e. correspond to the requirement that the tangent of one circle drawn at intersection point goes through the center of the other circle). This construction proves that the given data are not sufficient to determine unambiguously the radii of the circles.
