Tangent to a Circle
A tangent to a circle is a line that touches the circle at exactly one point (the point of tangency) and is perpendicular to the radius at that point.
Formula
\text{tangent length from external point } P: t = \sqrt{d^2 - r^2}
Definition
A tangent to a circle is a line that touches it at exactly one point, the point of tangency, without crossing through; it is always perpendicular to the radius drawn to that point. From an external point $P$ at distance $d$ from the center, the tangent length is $t=\sqrt{d^2-r^2}$, and two tangents drawn from the same external point are always equal in length, a fact used to prove that a circle can be inscribed in a triangle. Formally, the tangent at point $T$ is the limit of secant lines through $T$ as the second intersection point approaches $T$; for the unit circle, the tangent at $(\cos\theta,\sin\theta)$ has equation $x\cos\theta+y\sin\theta=1$.
Example
A ball resting on a floor touches it tangentially at one point; the word "tangent" comes from Latin "tangere," to touch. For external point $P$ at distance $13$ from center $O$, radius $5$: tangent length $=\sqrt{169-25}=\sqrt{144}=12$, forming a $5$-$12$-$13$ right triangle with $PO$. For circle $x^2+y^2=25$ at point $(3,4)$: tangent equation $3x+4y=25$, with radius direction $(3,4)$ and tangent direction $(-4,3)$ giving dot product $-12+12=0$, confirming perpendicularity.
Key Insight
The perpendicularity between a tangent and its radius is the key property used to solve tangent problems and construct tangent lines by hand. The equal-tangent-lengths property from any external point is what makes it possible to inscribe a circle inside a triangle, tangent to all three sides, and the tangent line as the limit of secant lines is precisely the geometric definition of the derivative, the slope of the tangent equals $dy/dx$ at that point, a connection both Leibniz and Newton used to motivate their definitions of calculus.