External Tangent Congruence Theorem

Consider two triangles.

• The triangle formed by the radius $\overline{OB},$ the segment $\overline{OA},$ and the tangent segment $\overline{AB}.$
• The triangle formed by the radius $\overline{OC},$ the segment $\overline{OA},$ and the tangent segment $\overline{AC}.$

These two triangles can be visualized in the diagram.

Note that $B$ and $C$ are points of tangency. Therefore, by the Tangent to Circle Theorem, $\angle B$ and $\angle C$ are right angles. Consequently, $△ABO$ and $△ACO$ are right triangles.

Because all radii of the same circle are congruent, it can be said that $\overline{OB}$ and $\overline{OC}$ are congruent. Moreover, $△ABO$ and $△ACO$ share the same hypotenuse $\overline{OA}.$ By the Reflexive Property of Congruence, $\overline{OA}$ is congruent to itself.

Combining all of this information, it can be said that the hypotenuse and one leg of $△ABO$ are congruent to the hypotenuse and the corresponding leg of $△ACO.$

Therefore, by the Hypotenuse-Leg Theorem, $△ABO$ and $△ACO$ are congruent triangles. Since corresponding parts of congruent figures are congruent, it can be said that $\overline{AB}$ and $\overline{AC}$ are congruent.