Thales' Theorem/Proof 1

Proof

 * Thales theorem.jpg

Let $O$ be the center of the circle, and define the vectors:
 * $\mathbf u = \overrightarrow{OC}$
 * $\mathbf v = \overrightarrow{OB}$
 * $\mathbf w = \overrightarrow{OA}$

Thus:
 * $\overrightarrow{AC} = \mathbf u - \mathbf w$
 * $\overrightarrow{BC} = \mathbf u - \mathbf v$

Then:

From Non-Zero Vectors Orthogonal iff Perpendicular, it follows that $AC$ and $BC$ are perpendicular to each other.