Why the tangent line has two equivalent definitions,looking for a long time can not find

Question

Why the tangent line has two equivalent definitions, one is according to the Angle, the other is according to the limit position of the cut line, how the two definitions are equivalent, please give proof, looking for a long time can not find

Answer 1

For this, I will assume that you're talking about a circle, since a tangent to a circle is (i) perpendicular to the radius at the point of tangency and (ii) intersects the circumference at exactly one point (limit of a secant).

In the diagram, $A$ is the centre of the circle and $B$ is the point of tangency. Let $C$ be any point on the tangent passing through $B$ such that $B\not=C$. Since the shortest distance from a point ($A$) to a line (the tangent at $B$) is the perpendicular onto that line ($\overline{AB}$), we must have that the tangent is perpendicular to the radius $\overline{AB}$.