Radical axis

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Definitions

The power of point $A$ with respect to circle $\omega$ (with radius $r$ and center $O$ , which shall thereafter be dubbed $pow(P, \omega)$ , is defined to equal $OP^2 - r^2$ .

The radical axis of two circles $\omega_1, \omega_2$ is defined as the locus of the points $P$ such that the power of $P$ with respect to $\omega_1$ and $\omega_2$ are equal. In other words, if $O_i, r_i$ are the center and radius of $\omega_i$ , then a point $P$ is on the radical axis if and only if $PO_1^2 - r_1^2 = PO_2^2 - r_2^2$

Results

Theorem 1: (Power of a Point) If a line drawn through point P intersects circle $\omega$ at points A and B, then $pow(P, \omega) = PA * PB$ . Theorem 2: (Radical Axis Theorem)

a. The radical axis is a line perpendicular to the line connecting the circles' centers (line $l$ ).

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 $l$ through point A, the point on $l$ such that $pow(A, \omega_1) = pow(A, \omega_2)$ .

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.

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Radical axis

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