### Circle Inscribed in a Circular Segment: What is this about?

A Mathematical Droodle

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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.

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

- Angle Preservation Property
- Apollonian Circles Theorem
- Archimedes' Twin Circles and a Brother
- Bisectal Circle
- Chain of Inscribed Circles
- Circle Inscribed in a Circular Segment
- Circle Inversion: Reflection in a Circle
- Circle Inversion Tool
- Feuerbach's Theorem: a Proof
- Four Touching Circles
- Hart's Inversor
- Inversion in the Incircle
- Inversion with a Negative Power
- Miquel's Theorem for Circles
- Peaucellier Linkage
- Polar Circle
- Poles and Polars
- Ptolemy by Inversion
- Radical Axis of Circles Inscribed in a Circular Segment
- Steiner's porism
- Stereographic Projection and Inversion
- Tangent Circles and an Isosceles Triangle
- Tangent Circles and an Isosceles Triangle II
- Three Tangents, Three Secants
- Viviani by Inversion
- Simultaneous Diameters in Concurrent Circles
- An Euclidean Construction with Inversion
- Construction and Properties of Mixtilinear Incircles
- Two Quadruplets of Concyclic Points
- Seven and the Eighth Circle Theorem
- Invert Two Circles Into Equal Ones

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