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
Copyright © 1996-2007 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
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1/(R - d)2 + 1/(R + d)2 = 1/r2,
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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:
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 BAI + ICB = 90°.
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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.
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Also, the Pythagorean theorem applied to that triangle gives
| (2) |
(AK + CL)² = AI² + CI².
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From (1) and (2),
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r²·(AI² + CI²) = AI² · CI²,
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or,
| (3) |
1/r² = 1/AI² + 1/CI².
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Let AI and CI produced intersect O(R) in F and E. Then EF is a diameter of C(R) because
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| DOF + DOE | = 2(DAF + DCE) |
| | = BAD + BCD |
| | = 180°. |
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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). |
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Considering the diameter of O(R) through I, the intersecting chords theorem gives
| (5) |
AI · FI = CI · EI = R² - d²
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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. |
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Finally, from (3) and (6),
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| 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. |
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References
- N. Altshiller-Court, 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. 107-110
- 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. 837-839
- J. C. Salazar, Fuss' Theorem, The Mathematical Gazette, v 90, n 518 (July 2006), pp. 306-307.
Poncelet Porism
Copyright © 1996-2008 Alexander Bogomolny
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