Tangents from an external point

From a point external to a circle exactly two tangents can be drawn, and the two tangent segments enjoy a notable property: they are congruent.

Theorem — Two tangents from an external point

From a point $P$ external to a circle, exactly two tangents can be drawn. The tangent segments $PA$ and $PB$ are congruent: $PA\cong PB$.

The radii $OA$ and $OB$ are perpendicular to the tangents at the points of tangency; $OP$ is the common hypotenuse of the two right-angled triangles.

Proof — Congruent tangent segments

1. Hypothesis: $PA$ and $PB$ tangent, so $OA\perp PA$ and $OB\perp PB$, with $OA\cong OB$ (radii).
2. We join $O$ to $P$.
3. We examine the right-angled triangles $PAO$ and $PBO$:
   • $OA\cong OB$ (radii)
   • $OP$ common hypotenuse
   • $\widehat{OAP}=\widehat{OBP}=90°$
4. By the hypotenuse-leg criterion: $\triangle PAO\cong\triangle PBO$, hence $PA\cong PB$.

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Links

Topics: Euclidean circle
Concepts: Tangent
Methods: Circle tangents
Skills: Proving · Synthetic geometry