The Circle of Apollonius
The locus of a point C whose distance from a fixed point A is
a multiple of its distance from another fixed point B. If the
multiple is equal to 1, then the locus is a line -- the perpendicular
bisector of the segment AB. If the multiple is not equal to 1,
then it is a circle. The locus is called the Circle of Apollonius.
. 
Prove:
Given two fixed points A and B, the locus of point C such that
the ratio of AC to BC is constant is a circle.
Hint: See the Bisectors of
an Angle
Clearly, there is a Circle of Apollonius for each vertex of
triangle ABC.
Note that in an Isosceles triangle, the Circle of Apollonius
for the non-base angle becomes a perpendicular bisector to the
base. What happens for an equilateral triangle?
Prove:
The three Circles of Apollonius for a triangle ABC are concurrent
in two points.
Explore:
Add the circumcircle of triangle ABC to the figure of the triangle
and its the Circles of Apollonius. Click here
for a GSP sketch.
Prove:
Each of the three Circles of Apollonius in a given triangle ABC
is orthogonal to the circumcircle.
Hint: Consider one Circle of Apollonius and the circumcircle of
triangle ABC. Click here
for a GSP Sketch.
Conjecture?
Proof?
For reference, in this picture, the green, dark blue, and red
circles are the Circles of Apollonius for the triangle (heavy
black segments). The light blue circle is the circumcircle.