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A bicentric polygon is a polygon which has both an inscribed circle and a circumscribed one. For given two circles, the necessary and sufficient condition for existence of a bicentric triangle or quadrilateral is known as Chapple’s formula or Fuss’ formula, respectively. In this paper, we give natural extensions of these formulae. For an ellipse $$ E $$ E , the following two conditions are equivalent: (1) there exists a triangle and a quadrilateral, which $$ E $$ E is inscribed in and the unit circle is circumscribed about, respectively; (2) for some $$ a,b $$ a , b in the unit disk, $$ E $$ E is defined by the equation $$\begin{aligned} |z-a|+|z-b|=|1-\overline{a}b| \end{aligned}$$ | z - a | + | z - b | = | 1 - a ¯ b | or $$\begin{aligned} |z-a|+|z-b|=|1-\overline{a}b| \sqrt{(|a|^2+|b|^2-2)/(|a|^2|b|^2-1)}, \end{aligned}$$ | z - a | + | z - b | = | 1 - a ¯ b | ( | a | 2 + | b | 2 - 2 ) / ( | a | 2 | b | 2 - 1 ) , respectively. To prove them, we use some geometrical properties of Blaschke products on the unit disk, which also give new proofs of Chapple’s and Fuss’ formulae.
Computational Methods and Function Theory – Springer Journals
Published: Sep 26, 2013
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