Carlitz identity

Tendon tangent square

The Carlitz identity , according to Leonard Carlitz (1907–1999), is like Fuss's theorem an extension of Euler's theorem for triangles to tendon tangent quadrilaterals, and provides a formula for the distance between the centers of the circumference and the inscribed circle of a tendon tangent quadrilateral.

Designates the distance between the two center points, the radius of the perimeter and the radius of the inscribed circle, then the following equation applies ${\displaystyle x}$ ${\displaystyle R}$ ${\displaystyle r}$

${\displaystyle \displaystyle x^{2}=R^{2}-2Rr\cdot \mu }$

The factor is defined as follows: ${\displaystyle \mu }$

${\displaystyle \displaystyle {\begin{aligned}\mu &={\sqrt {\frac {(ab+cd)(ad+bc)}{(a+c)^{2}(ac+bd)}}}={\sqrt {\frac {(ab+cd)(ad+bc)}{(b+d)^{2}(ac+bd)}}}\\&={\frac {\cos({\tfrac {\alpha +\gamma }{4}})}{\cos({\tfrac {\alpha -\gamma }{4}})}}={\frac {\cos({\tfrac {\beta +\delta }{4}})}{\cos({\tfrac {\beta -\delta }{4}})}}\end{aligned}}}$

The sides (lengths) of the quadrilateral are with , , and referred to and , , and are the respective central angle of these sites. ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle c}$ ${\displaystyle d}$ ${\displaystyle \alpha }$ ${\displaystyle \beta }$ ${\displaystyle \gamma }$ ${\displaystyle \delta }$

In contrast to the equation provided by Fuss' theorem, the Carlitz identity corresponds exactly to the equation from Euler's theorem, except for the correction term . The proof of the Carlitz identity also provides the inequality , which is also referred to as the Carlitz inequality . This is also a direct consequence of Fuss' theorem. ${\displaystyle \mu }$ ${\displaystyle R\geq {\sqrt {2}}r}$

literature

• L. Carlitz: A Note on Circumscriptible Cyclic Quadrilaterals . Mathematics Magazine, Volume 38, No. 1 (January, 1965), pp. 33-35 ( JSTOR 2688014 )
• Albrecht Hess: Bicentric Quadrilaterals through Inversion . Forum Geometricorum, Volume 13 (2013), ISSN  1534-1178 , pp. 11-15