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].

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

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:

Also, the Pythagorean theorem applied to that triangle gives

From (1) and (2),

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):

Considering the diameter of O(R) through I, the intersecting chords theorem gives

It follows (from (4) and (5)) that

Finally, from (3) and (6),

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

• Poncelet's Porism
• Extouch Triangle in Poncelet Porism
• Fuss' Theorem
• Intouch Triangle in Poncelet Porism
• Euler's Formula and Poncelet Porism
• Poncelet Porism in Ellipses

Copyright © 1996-2009 Alexander Bogomolny