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.

Since there are no constraints given except the two radii, if we find one working example quadrilateral, we need only use this one to measure the distance.

There is a subset of of q's that can have an inclrcle (**ic**).

Likewise there is a subset of q's that can have a circumcircle (**cc**).

We look in the intersection of those subsets.

E.g, squares are candidates, but the ratio 12/7 does not equal sqrt(2).

Looking at more members found in the intersection: all rombi have an **ic,** but non-square rombi *cannot* have a **cc**! This brings us back to squares.

(It is mentioned in subsequent comments that some isosceles trapezoids are found in the intersection.)

Things get better with the kites, a superset of rombi. All kites have a **cc**, and the **right kites** (kites having a pair of opposed right angles) have a **cc**, so that's a good place to look to satisfy the 12/7 constraint. The right **right kite** will also have the convenience of bilateral symmetry, with the distance to be measured lying along the fold.

*Edited on ***October 27, 2019, 10:03 am**