Purser's Theorem

Let $t$, $u$, and $v$ be the lengths of the tangents to a Circle $C$ from the vertices of a Triangle with sides of lengths $a$, $b$, and $c$. Then the condition that $C$ is tangent to the Circumcircle of the Triangle is that

$$\pm at\pm bu\pm cv=0.$$

The theorem was discovered by Casey prior to Purser's independent discovery.

See also Casey's Theorem, Circumcircle