Euler's equation

One of the many results of Leonhard Euler in the elementary square geometry is related to the problem, when in the Euclidean plane to two given nested circles a convex exists quadrilateral both cyclic quadrilateral of the larger circle and tangential quadrilateral is the smaller circle. Euler found an equation for this , which is closely related to that in his theorem on the distance between the center of the circle and the center of a plane triangle . Euler's secretary Nikolaus Fuß provided the first published presentation and derivation of the equation in 1798.

Representation of the equation

Euler's equation gives a convex square

The following theorem applies to the Euler-Fuß equation, which combines the corresponding theorem of Fuss and its inverse:

Given are two positive numbers and and two circles and the Euclidean plane , where have the radius and the radius . ${\displaystyle r}$ ${\displaystyle R}$ ${\displaystyle {\mathcal {K}}_{r}}$ ${\displaystyle {\mathcal {K}}_{R}}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle {\mathcal {K}}_{R}}$ ${\displaystyle R}$ ${\displaystyle {\mathcal {K}}_{r}}$ ${\displaystyle r}$

The lay disc from within the circular disk of , and it was . ${\displaystyle \operatorname {conv} ({\mathcal {K}}_{r})}$ ${\displaystyle {\mathcal {K}}_{r}}$ ${\displaystyle \operatorname {conv} ({\mathcal {K}}_{R})}$ ${\displaystyle {\mathcal {K}}_{R}}$ ${\displaystyle r<R}$

The distance of the two circle centers is . ${\displaystyle d}$

Then:

Then and only then does a convex square exist in the Euclidean plane with as an inscribed circle and as a circumference , if the equation ${\displaystyle {\mathcal {K}}_{r}}$ ${\displaystyle {\mathcal {K}}_{R}}$

${\displaystyle {\frac {1}{(R+d)^{2}}}+{\frac {1}{(R-d)^{2}}}={\frac {1}{r^{2}}}}$

is satisfied.

Remarks

• In Heinrich Dörrie's Mathematical Miniatures , the Euler-Fuß equation is also called the square formula under the heading of Fuß . Dörrie gives - using other parameters - the following equation:

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

• A convex square, which has both a circumference and an inscribed circle, is also called a bicentric square , according to Heinrich Dörrie .
• In his triumph of mathematics , Heinrich Dörrie points out that Nikolaus Fuß also found the corresponding formulas for the bicentric pentagon , hexagon , heptagon and octagon .

Sources and literature

• Julian Lowell Coolidge : A Treatise on the Circle and the Sphere (archive.org) . (Corrected reprint of the 1916 edition). Chelsea Publishing Company , Bronx, NY 1971, ISBN 0-8284-0236-1 .
• Heinrich Dörrie: triumph of mathematics . 100 famous problems from two millennia of mathematical culture. 5th edition. Physica-Verlag , Würzburg 1958.
• Heinrich Dörrie: Mathematical miniatures . 2nd Edition. Sendet, Wiesbaden 1979, ISBN 3-500-21150-X (unchanged reprint of the 1943 edition).
• Max Simon : About the development of elementary geometry in the XIX. Century . Annual report of the German Mathematicians Association. tape 15 . BG Teubner Verlag , Leipzig 1906.

Individual evidence

1. ^ Julian Lowell Coolidge: A Treatise on the Circle and the Sphere. 1916 (reprint 1971, 2004), p. 44 ff
2. ↑ Max Simon: About the development of elementary geometry in the XIX. Century. 1906, p. 108
3. ↑ ^{a b} Heinrich Dörrie: Mathematical miniatures. 1979, pp. 71-72, 115
4. ↑ Julian Lowell Coolidge: op. Cit. Pp. 46 ff , 117-118
5. ↑ ^{a b} Dörrie, op.cit., P. 522
6. ^ Heinrich Dörrie: Triumph of mathematics. 1958, p. 196