Problem — Tangent segments from an external point

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From a point $P$ outside a circle with centre $O$, draw the two tangents, which touch the circle at the points $A$ and $B$. Prove that the two tangent segments $PA$ and $PB$ are congruent.

Solution

The radii $OA$ and $OB$ are perpendicular to the respective tangents, so the triangles $OAP$ and $OBP$ are right-angled at $A$ and $B$ respectively. They share the hypotenuse $OP$ and have legs $\overline{OA} = \overline{OB} = r$ (radii). By the congruence criterion for right triangles (hypotenuse and one leg congruent) the two triangles are congruent. Consequently $\overline{PA} = \overline{PB}$. $\blacksquare$