Thales' Theorem/Proof 1

Theorem
Let $A$ and $B$ be two points on opposite ends of the diameter of a circle.

Let $C$ be another point on the circle such that $C \ne A, B$.

Then the lines $AC$ and $BC$ are perpendicular to each other.

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 Dot Product of Perpendicular Vectors, it follows that $AC$ and $BC$ are perpendicular to each other.

Legend has it that he sacrificed an ox in honour of the discovery.

On the other hand, some attribute this theorem to Pythagoras.