Power of Point Special Cases

Question

Does power of point work if one of the line is tangent to the circumference of the circle and the other goes through the center. The distance between the point of intersection between the lines and the closet point on the circle is known. Can I use power of point?

Answer 1

There is no special case. The power of a point $P$ with respect to a circle with radius $R$ and centre $O$ is simply $OP^2-R^2$. If $P$ lies inside the circle and $AB$ is a chord of that circle through $P$, then $$ OP^2 - R^2 = - PA\cdot PB =\text{pow}_{\Gamma}(P)<0$$ and if $P$ lies outside the circle and a line through $P$ meets the circle at $A$ and $B$, $$ OP^2- R^2 = PA\cdot PB = \text{pow}_{\Gamma}(P) > 0.$$