Question

prove that tangents drawn at the ends of a diameter of a circle are parallel

Answer 1

Question #38228, Math, Other

Prove that tangents drawn at the ends of a diameter of a circle are parallel.

Solution

Let $AB$ be the diameter of a circle with the center $O$, the tangent line $CD$ have the tangency point $A$ and the tangent line $EF$ have the tangency point $B$ (see the figure).

Since the tangent $CD$ has only one common point $A$ with the circle, then $OA$ is the shortest distance from the center $O$ to any point of the tangent $CD$, and hence the diameter $AB$ is perpendicular to the line $CD$ and vice versa. Similarly, the tangent line $EF$ is perpendicular to the diameter $AB$. If we assume that these two tangents intersect at some point $P$, then two perpendiculars to the line $AB$ will be dropped from this point, which is impossible. Therefore, we were wrong to assume there was a point of intersection. The tangents do not intersect – they are parallel.