Power of a Point Theorem
Topic Index | Geometry Index | Regents Exam Prep Center

Something to explore with your students ....

In 1826, Jacob Steiner used the word "power" to refer to the relationship of a point to two intersecting lines and a circle. 

Power of a Point Theorem:

Given circle O, point P not on the circle, and a line through P intersecting the circle in two points.  The product of the length from P to the first point of intersection and the length from P to the second point of intersection is constant for any choice of a line through P that intersects the circle.  This constant is called the "power of point P".
 


If P is outside the circle ....


This becomes the theorem we know as the theorem of intersecting secants.

 If P is inside the circle ....

This becomes the theorem we know as the theorem of intersecting chords.

Special Cases:

Should one of the lines be tangent to the circle, point A will coincide with point D, and the theorem still applies.



This becomes the theorem we know as the theorem of secant-tangent theorem.

 Should both of the lines be tangents to the circle, point A coincides with point D, point C coincides with point B, and the theorem still applies.



This becomes the theorem we know as the theorem of two tangents.



The proof of this theorem uses similar triangles.....

1.  1.  Given
2. 2.  Two points determine exactly one line.
3.  3.  Reflexive Property (Identity)
4.  4.  In a circle, the measure of an inscribed angle is one-half the measure of its intercepted arc.
5.  5.  Substitution (or Transitive)
6.  6.  Congruent angles are angles of equal measure.
7.  7.  AA  (If two angles of one triangle are congruent to the corresponding angles of another triangle, the triangles are similar.
8.  8.  Corresponding sides of similar triangles are in proportion.
9.  9.  In a proportion, the product of the means equals the product of the extremes.