Utilized Methods

Power of a point to a circle

Benjamin Florián

In geometry, the power of a point to a circle is a mathematical concept that reflects the relative distance of a given point to a circle. The most common sign for it is m with a parenthesis, where the name of the point is given first, followed by the name of the circle (e.g. m(M,k)). It is most often used in problems where it is necessary to work with secants and tangents of circles. Namely, the power of a point to a circle can be used as a solution to the Apollonian problem of two points and a line, where the points are located on the same half-plane.

The power of a point to a circle is defined by: m(M,k)=|MS|2-r2 where M is a given point and k is a given circle of radius r and center S.

With this definition we can already determine the position of the point in relation to the cirle.

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

From this relation, another definition of the power of a point to a circle can be expressed using points on the circle, which with the given point form a secant on the circle, that passes through its center.

If the point M is located outside the circle, on which the points A and B lie, it is true, that |MA|=|MS|+r and also that |MB|=|MS|-r.

If we transform the original definition m(M,k)=|MS|2-r2 into (|MS|+r)·(|MS|-r), we can plug in the relations for the segments |MA| and |MB|, resulting in the relation: m(M,k)=|MA|·|MB|

In the case of the point M lying inside the circle, it is true for the segments that |MA|=r+|MS| and |MB|=r-|MS|. Therefore, we rewrite the first definition as m(M,k)=-(r2-|MS|2) and after substitution we get the relation: m(M,k)=-|MA|·|MB|

However, this relation is insufficient, since it is not clear, whether it also holds true for segments that do not pass through the center of the circle. It can be proven by using two more points that lie on the circle.

From the above construction, we can see that the triangles MAD and MBC are similar since they have the same two interior angles. Consequently, according to the relation for similarity of triangles, we can deduce the relation |MA|∶|MC|=|MD|∶|MB| and therefore |MA|·|MB|=|MD|·|MC|. Therefore, we know that: m(M,k)=|MA|·|MB|=|MD|·|MC|

This proves that the previous relation holds true for all secants of the circle k passing through the point M.

The last definition of a point to a circle is demonstrated in the case where a tangent of the circle k passes through the point M.

We can see from the construction, that |ST|=r and that it forms a right triangle with the segment |MS|. We can then use the Pythagorean theorem to derive a relation for |MT|: |MT|2=|MS|2-r^2

Finally, we just plug the first definition of the power point into the relation, which gives us: m(M,k)=|MT|^2

To summarize all the relations in one at the end, the power of a point to a circle is defined by the following relation: m(M,k)=|MA|-|MB|=|MD|-|MC|=|MT|^2