Radical axis
Work in Progress
Contents
Definitions
The power of point with respect to circle (with radius and center , which shall thereafter be dubbed , is defined to equal .
The radical axis of two circles is defined as the locus of the points such that the power of with respect to and are equal. In other words, if are the center and radius of , then a point is on the radical axis if and only if
Results
Theorem 1: (Power of a Point) If a line drawn through point P intersects circle at points A and B, then . Theorem 2: (Radical Axis Theorem)
a. The radical axis is a line perpendicular to the line connecting the circles' centers (line ).
b. If the two circles intersect at two common points, their radical axis is the line through these two points.
c. If they intersect at one point, their radical axis is the common internal tangent.
d. If the circles do not intersect, their radical axis is the perpendicular to through point A, the point on such that .
Theorem 3: (Radical Axis Concurrence Theorem) The three pairwise radical axes of three circles concur at a point, called the radical center.
Proofs
Theorem 1 is trivial Power of a Point, and thus is left to the reader as an exercise. (Hint: Draw a line through P and the center.)
Theorem 2 shall be proved here. Assume the circles are and with centers and and radii and , respectively. (It may be a good idea for you to draw some circles here.)
Lemma 1: Let be a point in the plane, and let be the foot of the perpendicular from to . Then .
The proof of the lemma is an easy application of the Pythagorean Theorem and will again be left to the reader as an exercise.
Lemma 2: There is an unique point P on line such that .
Proof: First show that P lies between and . Then, use the fact that (a constant) to prove the lemma.
Lemma 1 shows that every point on the plane can be equivalently mapped to a line on . Lemma 2 shows that only one point in this mapping satisfies the given condition. Combining these two lemmas shows that the radical axis is a line, completing part (a).
Parts (b), (c), and (d) will be left to the reader as an exercise.
Now, try to prove Theorem 3 on your own!
Exercises
If you haven't already done so, prove the theorems and lemmas outlined in the proofs section. If you have done so, prove them again. (Repetition is the best way to master a new concept.)
Problem 1. See 2009 USAMO Problem 1.