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].
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