Fuss' Theorem
The applet below illustrates Poncelet's porism for quadrilaterals. It is based on a formula by Nicolaus Fuss (17551826), a student and friend of L. Euler. Fuss also found the corresponding formulas for the bicentric pentagon, hexagon, heptagon, and octagon [Dörrie, p. 192].


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Copyright © 19962007 Alexander Bogomolny
The circumradius (R), the inradius (r) and the distance between the circumcenter and the incenter (d) of a bicentric quadrilateral stand in an elegant relationship

1/(R  d)^{2} + 1/(R + d)^{2} = 1/r^{2},

It looks very much the same as Euler's formula for triangles, except for the exponent of 2.
Here's a very short proof of this fact by J. C. Salazar:
Let the two circles be O(R) and I(r), K and L be the points where circle I(r) touches AB and BC.
Since quadrilateral ABCD is cyclic the angles at A and C are supplementary making angles BAI and ICB complimentary:
By construction, IK = IL = r. Two triangles AIK and CIL combined form a right triangle with legs AI and CI and the hypotenuse AK + CL. Its area can be found in two ways:
(1) 
r·(AK + CL) = AI · CI.

Also, the Pythagorean theorem applied to that triangle gives
(2) 
(AK + CL)² = AI² + CI².

From (1) and (2),

r²·(AI² + CI²) = AI² · CI²,

or,
(3) 
1/r² = 1/AI² + 1/CI².

Let AI and CI produced intersect O(R) in F and E. Then EF is a diameter of C(R) because

∠DOF + ∠DOE  = 2(∠DAF + ∠DCE) 
 = ∠BAD + ∠BCD 
 = 180°. 

We are then in a position to apply the formula for the length of a median (IO) in a triangle (EFI):
(4) 
EI² + FI²  = 2 IO² + EF² / 2 
 = 2 (d² + R²). 

Considering the diameter of O(R) through I, the intersecting chords theorem gives
(5) 
AI · FI = CI · EI = R²  d²

It follows (from (4) and (5)) that
(6) 
1/AI² + 1/CI²  = FI²/(R²  d²)² + EI²/(R²  d²)² 
 = (EI² + FI²) / (R²  d²)² 
 = 2 (R² + d²) / (R²  d²)². 

Finally, from (3) and (6),

1/r²  = 2 (R² + d²) / (R²  d²)² 
 = [(R + d)² + (R  d)²)] / (R²  d²)² 
 = 1/(R + d)² + 1/(R  d)². 

References
 N. AltshillerCourt, College Geometry, Dover, 1980
 J. Casey, A Sequel to Euclid, Scholarly Publishing Office, University of Michigan Library (December 20, 2005), reprint of the 1888 edition, pp. 107110
 J. L. Coolidge, A Treatise On the Circle and the Sphere, AMS  Chelsea Publishing, 1971, p. 45
 H. Dörrie, 100 Great Problems Of Elementary Mathematics, Dover Publications, NY,1965
 F. G.M., Exercices de Géométrie, Éditions Jacques Gabay, sixiéme édition, 1991, pp. 837839
 J. C. Salazar, Fuss' Theorem, The Mathematical Gazette, v 90, n 518 (July 2006), pp. 306307.
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