Set of feet

The set of feet , named after Nikolaus Fuss (1755–1826), provides a formula for the distance between the center of the inscribed circle and the center of the circumference of a square tendon tangent.

{\ displaystyle x = {\ sqrt {R ^ {2} + r ^ {2} -r {\ sqrt {4R ^ {2} + r ^ {2}}}}}}

Here refers to the distance between the two centers, the radius of the circumscribed circle and the radius of the inscribed circle. However, the sentence is usually not presented as an explicit distance formula, but as a fraction equation that relates the three quantities to one another. ${\ displaystyle x}$ ${\ displaystyle R}$ ${\ displaystyle r}$

{\ displaystyle {\ frac {1} {(Rx) ^ {2}}} + {\ frac {1} {(R + x) ^ {2}}} = {\ frac {1} {r ^ {2 }}}}

Another alternative representation as an equation without fractions is:

{\ displaystyle \ displaystyle 2r ^ {2} (R ^ {2} + x ^ {2}) = (R ^ {2} -x ^ {2}) ^ {2}}

With the sentence named after him, Fuss, who was Euler's secretary in Saint Petersburg , transferred a corresponding sentence from Euler about the distance between the center points of the circumference and the incircle of triangles to tendon tangent quadrilaterals. With the Carlitz identity there is another formula for the distance between the center points, which, however, not only requires the two radii, but also the lengths of the sides of the square of the tendon tangents.

If one takes into account that in the last of the three forms of representation above , one obtains and thus an inequality for the two radii, which can be understood as an analogue to Euler's inequality in the triangle. ${\ displaystyle x ^ {2}> 0}$ ${\ displaystyle 2r ^ {2} (R ^ {2}) <(R ^ {2}) ^ {2}}$ ${\ displaystyle {\ sqrt {2}} r <R}$

The converse also applies, that is, if two circles fulfill the above equations, there is a tendon tangent quadrilateral that has the circles as a circumference and an inscribed circle. Because of Poncelet's closure theorem, there is always an infinite number of tendon tangent quadrilaterals with this property.

literature

WE Byerly: The In- and-Circumscribed Quadrilateral . Annals of Mathematics, Second Series, Vol. 10, No. 3 (Apr., 1909), pp. 123-128 ( JSTOR 1967103 )
Juan Carlos Salazar: 90.46 feet 'theorem . The Mathematical Gazette, Volume 90, No. 518 (July, 2006), pp. 306-307 ( JSTOR 1967103 )
Heinrich Dörrie: 100 Great Problems of Elementary Mathematics . Dover Publications, New York 1965, ISBN 0-486-61348-8 , pp. 188-193
Claudi Alsina, Roger B. Nelsen: Pearls of Mathematics: 20 geometric figures as starting points for mathematical exploratory trips. Springer, 2015, ISBN 9783662454619 , p. 123 ( excerpt from the English edition (Google) )
Albrecht Hess: Bicentric Quadrilaterals through Inversion . Forum Geometricorum, Volume 13 (2013), ISSN 1534-1178, pp. 11-15

Web links

Fuss' theorem on cut-the-knot.org
Paul Yiu: Euclidean Geometry Notes . Script, Florida Atlantic University, pp. 159-162