The Complex Geometry of Blaschke Products of Degree 3 and Associated Ellipses

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 bicentric triangle for these two circles is known as Chapple's formula or Euler's theorem.

As one of natural extensions of this formula, we characterize the inscribed ellipses of a triangle which is inscribed in the unit circle.  We also discuss the condition for the "circumscribed" ellipse of a triangle which is circumscribed about the unit circle.

For the proof of these results, we use some geometrical properties of Blaschke products on the unit disk.