Fuss' TheoremThe 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].
|Activities| |Contact| |Front page| |Contents| |Geometry| |Store| Copyright © 1996-2007 Alexander BogomolnyThe 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:
By construction, IK = IL = r. Two triangles AIK and CIL combined form a right triangle with legs AI and CI and the hypotenuse
Also, the Pythagorean theorem applied to that triangle gives
or,
Let AI and CI produced intersect O(R) in F and E. Then EF is a diameter of C(R) because
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
References
Poncelet Porism
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