What Is the Power of a Point Theorem?

You can use the power of a point to find lengths and angles of geometric figures that involve circles.

This formula comes from the fact that the triangles $△PBD$ and $△PCA$ are similar. They are similar because they have two equal angles, which means the third one is also the same size. Both triangles share the angle $\angle P$, and $\angle PBD=\angle PCA$ because they span the same circular sector, the sector between $A$ and $D$. That means you can use the ratio between corresponding sides to find the formula above:

$$\begin{array}{llllllll}\hfill \frac{PB}{PC}& =\frac{PD}{PA}& \hfill & |\cdot PA& \hfill & & \hfill & \\ \hfill \frac{PA\cdot PB}{PC}& =PD& \hfill & |\cdot PC& \hfill & & \hfill & \\ \hfill PA\cdot PB& =PD\cdot PC& \hfill & & \hfill & \end{array}$$

Some lines intersect the circle at only one point—its tangents. If one of the lines is a tangent, the power of a point theorem becomes

$${(PA)}^2=PC\cdot PD.$$

This makes sense, because you can imagine that there are two intersections $A$ and $B$ that lie right on top of each other, and they have the exact same distance to $P$. In that case, $PB=PA$. If you insert that into the original formula, you see how the second formula appears:

$$\begin{array}{llll}\hfill PA\cdot PB& =PC\cdot PD& \hfill & \\ \hfill PA\cdot PA& =PC\cdot PD& \hfill & \\ \hfill {(PA)}^2& =PC\cdot PD& \hfill & \end{array}$$