Two tangents from an external point have equal length

Depends on: Tangent is perpendicular to the radius at the point of tangency, HL (hypotenuse-leg) congruence.

Statement

Let $\odot O$ be the circle of centre $O$ and radius $R$, and let $P$ be a point outside $\odot O$. From $P$ draw two tangents to $\odot O$, touching at $A$ and $B$. Then the two tangent lengths are equal:

Furthermore, the segment $OP$ joining the centre to $P$ bisects both symmetric angles:

That is, $OP$ is simultaneously the Angle bisector ⇔ equidistant from sides of $\angle APB$ and of $\angle AOB$.