# Inscribed Ellipses and Blaschke Products

Inscribed Ellipses and Blaschke Products 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

# Inscribed Ellipses and Blaschke Products

, Volume 13 (4) – Sep 26, 2013
17 pages

/lp/springer-journals/inscribed-ellipses-and-blaschke-products-OrOWc2RBaL
Publisher
Springer Journals
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-013-0037-8
Publisher site
See Article on Publisher Site

### Abstract

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.

### Journal

Computational Methods and Function TheorySpringer Journals

Published: Sep 26, 2013

### References

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