The principal control is the point of tangency of the red line

. Drag it anywhere on the black circle.

You can vary the central angle of the light red segment with the "segment angle" slider. The allowed range is limited to 0.2 to 6 radians.

You can choose the segment base with the "segment base" setter bar. If it is

, then the segment is included when you sweep a radius clockwise from

to

. If it is

, then the segment is included if you sweep a radius clockwise from

to

.

With the "vertex angle" slider, you can vary the angle of intersection of the lines on which

and

are found. It has a maximum value of 3 radians. If it is zero, the lines are parallel.

The segment theorem may be stated as follows. Given two lines

and

tangent to a given circle

, at point

on

construct a tangent to

that intersects

at

and

at

. Construct a circle through

and

such that the central angle of arc

is a predetermined angle. For any choice of

, this circle will be tangent to two fixed circles that are also tangent to both

and

. Note that there are other circles that are tangent to the red circle and lines

and

, but they are not fixed when

is moved.

In this Demonstration, circle

and lines

and

are black. The segment whose chord is tangent to

is light red, and the circle of this segment is red. The two fixed circles that are tangent to

and

and the red circle are light blue. The tangent point can be moved anywhere on

.

A proof of the segment theorem appears in [1]. It is used to prove, among other things, the Feuerbach theorem. The segment theorem is also used to prove the Feuerbach theorem in non-Euclidean space in [2].