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Fancy quadrilateral (Posted on 2019-10-25) Difficulty: 4 of 5
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.

No Solution Yet Submitted by Danish Ahmed Khan    
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Hints/Tips re: the right starting point? - kite solution | Comment 2 of 4 |
(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).

  Posted by Brian Smith on 2019-10-26 13:14:02
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