# Fuss' Theorem

The applet below illustrates Poncelet's porism for quadrilaterals. It is based on a formula by Nicolaus Fuss (1755-1826), 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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Explanation 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/r2,

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:

 ∠BAI + ∠ICB = 90°.

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

1. N. Altshiller-Court, College Geometry, Dover, 1980
2. J. Casey, A Sequel to Euclid, Scholarly Publishing Office, University of Michigan Library (December 20, 2005), reprint of the 1888 edition, pp. 107-110
3. J. L. Coolidge, A Treatise On the Circle and the Sphere, AMS - Chelsea Publishing, 1971, p. 45
4. H. Dörrie, 100 Great Problems Of Elementary Mathematics, Dover Publications, NY,1965
5. F. G.-M., Exercices de Géométrie, Éditions Jacques Gabay, sixiéme édition, 1991, pp. 837-839
6. J. C. Salazar, Fuss' Theorem, The Mathematical Gazette, v 90, n 518 (July 2006), pp. 306-307. #### Poncelet Porism • 