Tangent to a circle
= Tangent to a circle
At a point $T$ on a <circle> with centre $O$, the <tangent line> is perpendicular to the <radius> $OT$. For example, differentiating $|\mathbf r(s)-O|^2=R^2$ along a <parametric curve> on the <circle> gives $(\mathbf r-O)\cdot\mathbf r'=0$, showing that the radial <vector> and tangent direction are <orthogonal vectors>. Conversely, the perpendicular <straight line> through $T$ has only that point in common with the <circle>: for a nonzero displacement $v$ along it, $|T+v-O|^2=R^2+|v|^2>R^2$.