Power of a point to a circle

In geometry, the power of a point with respect to a circle is a real number that describes the position of the point relative to the circle. It is most commonly denoted by m. It is used especially in situations involving secants and tangents of circles. The power of a point can be used, for example, when solving the Apollonian problem defined by two points and a line, provided that both points lie in the same half-plane determined by the given line.

Let k be a circle with center S and radius r. The power of a point M with respect to the circle k is defined by the relation:

m(M,k)=|MS|²−r²

where M is the given point, k is the given circle, S is its center, and r is its radius.

The sign of the power tells us the position of the point relative to the circle.

• If m(M,k)>0, the point M lies outside the circle.
• If m(M,k)<0, the point M lies inside the circle.
• If m(M,k)=0, the point M lies on the circle.

From this relation, we can derive another expression for the power of a point using a secant passing through the point M. First, consider a secant that passes through both the point M and the center S. Let its intersection points with the circle be A and B.

If the point M lies outside the circle, then |MB|=|MS|−r a |MA|=|MS|+r.

The original definition

m(M,k)=|MS|²−r²

can be rewritten using the difference of squares:

m(M,k)=(|MS|−r)·(|MS|+r)

After substituting the relations for the segment lengths |MA| and |MB|, we obtain:

m(M,k)=|MA|·|MB|

If the point M lies inside the circle, then |MA|=r+|MS| and |MB|=r−|MS|. Therefore, the original definition can be rewritten as:

m(M,k)=|MS|²−r²=−(r²−|MS|²)

After substituting the relations for the segment lengths, we obtain:

m(M,k)=−|MA|·|MB|

So far, we have derived the relation only for a secant passing through the center of the circle. We will now verify that, for a point M lying outside the circle, the same relation also holds for any secant of the circle passing through the point M.

From the construction, we can see that the triangles MAD and MBC are similar because they have two equal interior angles. From the similarity of the triangles, we get the relation

|MA|∶|MC|=|MD|∶|MB|

and therefore

|MA|·|MB|=|MD|·|MC|.

Thus we obtain:

m(M,k)=|MA|·|MB|=|MD|·|MC|

This verifies that, for a point M lying outside the circle, the same relation holds for all secants of the circle k passing through the point M.

Another useful expression for the power of a point is obtained when a tangent to the circle k is drawn from the point M. Let the point of tangency be T.

From the construction, we can see that |ST|=r and that the segments MS, ST, and MT form a right triangle. By the Pythagorean theorem, we have:

|MT|²=|MS|²−r²

Comparing this relation with the definition of the power of a point, we obtain:

m(M,k)=|MT|²

For a point M lying outside the circle, the power of the point with respect to the circle can therefore be expressed by the following relations:

m(M,k)=|MS|²−r²=|MA|·|MB|=|MD|·|MC|=|MT|²

If the point M lies inside the circle, its power is negative. When working with segment lengths, it is therefore necessary to include a minus sign in this case.