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.