---
tag: 0QX7E9KAAX
kind: statement
role: theorem
name: Tangent–secant theorem
msc: [51M04]
uses: [0T7DX8NY1X, 060W5CBZP6, 0NPPGHQNXQ, 02E9VT1RVQ]
sources:
  - url: https://en.wikipedia.org/wiki/Tangent%E2%80%93secant_theorem
    locator: lead section
same_as:
  wikidata: Q2267175
  wikipedia: Tangent–secant theorem
---

## Statement

Let $C$ be a [[circle|060W5CBZP6]] in the Euclidean plane $\mathbb{R}^2$, and let $d$ be the [[Euclidean distance|0NPPGHQNXQ]]. Let $P$ be a point outside $C$.
Let a [[tangent line|0T7DX8NY1X]] 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|02E9VT1RVQ]] of $P$ with respect to $C$.
It is the limit case of the [[intersecting secants theorem|0J5XR7BW2S]] 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 $\sqrt{8}$.