The radii of the incircle and the circumcircle of a quadrilateral are 7 and 12. Find the distance between the centers of the two circles.

(In reply to

the right starting point? by Steven Lord)

A right kite is an easy configuration to work with for this problem.

Let AB be the main diagonal of the right kite, it is also a diameter of the circumcircle. Let C be one of the other points of the kite. Then ABC is a right triangle.

Let I be the incenter of the kite. It will lay on AB. Let T on AC and U on BC be the points of tangency that the incircle makes on those sides. Then AIT and IBU are right triangles similar to ABC.

Let R be the circumradius and r be the inradius and d be the length of the offset between the incenter and circumcenter. Then AB=2R, AI=R+x, IB=R-x, IT=r, and UI=r.

By similar triangles then BC=2R*r/(R+x) and AC=2R*r/(R-x). Then using the Pythagorean theorem (2R)^2 = (2R*r/(R+x))^2 + (2R*r/(R+x))^2.

Simplfying yields the formula __1/r^2 = 1/(R+x)^2 + 1/(R-x)^2__. Then substituting r=7 and R=12 and solving yields x=**3*sqrt(2)**.