The theory of radical axis is a priceless geometric tool that can solve formidable geometric problems fairly readily. Problems involving it can be found on many major math olympiad competitions, including the prestigious USAMO. Therefore, any aspiring math olympian should peruse this material carefully, as it may contain the keys to one's future success.
Not all theorems will be fully proven in this text. The objective of this document is to introduce you to some key concepts, and then give you a chance to derive some of the beautiful results on your own. In that way, you will understand and retain the information in here much more solidly. Finally, your newfound knowledge will be tested on a few challenging problems that are exemplary examples on how radical axis theory can be used and why it pertains to that situation. I hope after you read this text, you will become a better math student, armed with another tool to solve difficult problems. But, anyway, good luck.
The power of point with respect to circle (with radius and center ), which shall thereafter be dubbed , is defined to equal .
Note that the power of a point is negative if the point is inside the circle.
The radical axis of two non-concentric 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 (i.e., the radical axis is the line that one gets when you subtract the equations of two circles).
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, and if one does not fully contain the other, their radical axis is the perpendicular to through point A, the unique 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.
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.)
First, we tackle part (b). Suppose the circles intersect at points and and point P lies on . Then by Theorem 1 the powers of P with respect to both circles are equal to , and hence by transitive . Thus, if point P lies on , then the powers of P with respect to both circles are equal.
Now, we prove the inverse of the statement just proved; because the inverse is equivalent to the converse, the if and only if would then be proven. Suppose that P does not lie on . In particular, line does not intersect X. Then intersects circles and a second time at distinct points and , respectively. (If is tangent to , for example, we adopt the convention that ; similar conventions hold for . Power of a Point still holds in this case. Also, notice that and cannot both equal , as cannot be tangent to both circles.) Because is not equal to , so does not equal , and thus by Theorem 1 is not congruent to , as desired. This completes part (b).
For the remaining parts, we employ a lemma:
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 via proof by contradiction, by using a bit of inequality theory and the fact that . 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 perpendicular to , completing part (a).
Parts (c) and (d) will be left to the reader as an exercise. (Also, try proving part (b) solely using the lemmas.)
Now, try to prove Theorem 3 on your own! (Hint: Let P be the intersection of two of the radical axes.)
If you haven't already done so, prove the theorems and lemmas outlined in the proofs section. Note: No solutions will be provided to the following problems. If you are stuck, ask on the forum.
Problem 1. Two circles and intersect at and . Point is located on such that and
15 . If the radius of is , find the radius of . Note: An error in this problem previously rendered it unsolvable.
Problem 2. Solve 2009 USAMO Problem 1. If you already know how to solve it.
Problem 3. Two circles P and Q with radii 1 and 2, respectively, intersect at X and Y. Circle P is to the left of circle Q. Prove that point A is to the left of if and only if .
Problem 4. Solve 2012 USAJMO Problem 1.
Problem 5. Does Theorem 2 apply to circles in which one is contained inside the other? How about internally tangent circles? Concentric circles?
Problem 6. Construct the radical axis of two circles. What happens if one circle encloses the other?
Problem 7. Solve 1995 IMO Problem 1 in two different ways. Compare your solutions with the solutions provided.
Let triangle points and be given. Denote Prove that
WLOG, the order of the points is as shown on diagram.
2021 AIME Problem 13. Circles and with radii and , respectively, intersect at distinct points and . A third circle is externally tangent to both and . Suppose line intersects at two points and such that the measure of minor arc is . Find the distance between the centers of and .
2014 USAMO Problem 5. Let be a triangle with orthocenter and let be the second intersection of the circumcircle of triangle with the internal bisector of the angle . Let be the circumcenter of triangle and the orthocenter of triangle . Prove that the length of segment is equal to the circumradius of triangle .
2017 USAMO Problem 3. Let be a scalene triangle with circumcircle and incenter Ray meets at and again at the circle with diameter cuts again at Lines and meet at and is the midpoint of The circumcircles of and intersect at points and Prove that passes through the midpoint of either or