Problem — Tangents at the endpoints of a diameter

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Let $AB$ be a diameter of a circle with centre $O$. Draw the tangents to the circle at the points $A$ and $B$. Prove that these tangents are parallel.

Solution

The tangent at a point is perpendicular to the radius reaching that point. Hence the tangent at $A$ is perpendicular to the radius $OA$, and the tangent at $B$ is perpendicular to the radius $OB$. But the points $A$, $O$, $B$ are collinear, since $AB$ is a diameter: the radii $OA$ and $OB$ both lie on the line $AB$. Two lines perpendicular to the same line are parallel to each other; therefore the two tangents are parallel. $\blacksquare$