Tangent–secant theorem

Statement

Let $C$ be a circle in the Euclidean plane $\mathbb{R}^2$, and let $d$ be the Euclidean distance. Let $P$ be a point outside $C$. Let a tangent line through $P$ touch $C$ at $T$. Let a second line through $P$ meet $C$ at the two points $G_1$ and $G_2$. Then $$d(P, T)^2 = d(P, G_1)\, d(P, G_2).$$

Remarks

Both sides are equal to the power of $P$ with respect to $C$. It is the limit case of the intersecting secants theorem where one secant becomes a tangent. The theorem is Proposition 36 of Book III of Euclid's Elements.

Explanations

L1 human

From a point outside a circle, the square of the tangent length equals the product of the two distances along any secant. For the unit circle and $P = (3, 0)$, the secant along the $x$-axis meets the circle at distances $2$ and $4$. So the tangent length is $\sqrt{8}$.