Circle Inscribed in a Circular Segment: What is this about?
A Mathematical Droodle

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at, download and install Java VM and enjoy the applet.

What if applet does not run?


|Activities| |Contact| |Front page| |Contents| |Store| |Geometry|

Copyright © 1996-2017 Alexander Bogomolny

Circle Inscribed in a Circular Segment

The applet suggests the following statement:

A chord ST is drawn in a circle C with center O. A circle C' is inscribed into the circular segment thus obtained that touches the chord ST at the point A and the circle C at the point B. Let M denote the midpoint of the arc defined by ST that does not include B. Then A, B, and M are collinear.

Let O' be the center of the inscribed circle. Then ΔAO'B is isosceles. Extend AB beyond A and let it intersect the perpendicular OM to ST at point N. The two triangles AO'B and NOB are similar. Indeed they have a common angle at B and, since ON||O'A, their respective angles at O' and O are also equal. ΔNOB is therefore isosceles. OB = ON, which implies N = M.

Here's another way of looking at the configuration. The two circles are homothetic with center B. The lowest point A of the circle C' is mapped by that homothety onto the lowest point of the circle C, which is M. But any two points that correspond by a homothety lie on a line through the center of the homothety.

There are two more proofs that use the inversion transformation.

Inversion - Introduction

|Activities| |Contact| |Front page| |Contents| |Store| |Geometry|

Copyright © 1996-2017 Alexander Bogomolny


Search by google: