Segment Theorem

Drag the point of tangency of the red line anywhere on the black circle. The points and are on the other two lines tangent to this circle. The light red segment with chord has a fixed central angle. The segment's red circle is always tangent to each of the two light blue circles.


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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].
[1] V. Protasov, "The Feuerbach Theorem," Quantum, 10(2), Nov/Dec 1999 pp. 4–9.
[2] A. V. Akopyan, "On Some Classical Constructions Extended to Hyperbolic Geometry." (May 2011) arXiv:1105.2153v1.
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