The Circle of Apollonius
The locus of a point C whose distance from a fixed point A is a multiple r of its distance from another fixed point B. If the r is 1, then the locus is a line -- the perpendicular bisector of the segment AB. If the r is not equal to 1, then the locus is a circle. The locus is called the Circle of Apollonius.
Write a Geometer's Sketchpad sketch to generate the locus.
Who was Apollonius?
Pictures.
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Prove:
Given two fixed points A and B, the locus of point C such that the ratio of AC to BC is constant not equal to 1 is a circle.
Hint: See the Bisectors of an Angle
Alternative Description
Clearly, there is a Circle of Apollonius for each vertex of triangle ABC.
For any triangle ABC, there is a Circle of Apollonius associated with each vertex. It is found by constructing the internal and external angle bisectors for an angle and locating the intersection points on side opposite the angle. These intercepts of the angle bisectors determine the diameter of the Circle of Apollonius.
Further conjecture
Construct a Geometer's Sketchpad sketch showing the three Circles of Apollonius for a given triangle.
Conjecture?
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.
Explore:
Construct the line determined by the two intersection points of the Circles of Apollonius for triangle ABC. Does it pass through the circumcenter?
Conjecture?
Proof?
Explore:
Each of the Circles of Apollonius divides the opposite side of the triangle in an external point. Show that these three external points are colinear.
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.
Click here for a GSP sketch of the above configuration.
