proving Thales’ theorem with vectors

Let the radius of the circle be $r$ and $AB$ a diameter of the circle. We make the dot product calculation

$$\overrightarrow{PA}\cdot \overrightarrow{PB}=(\overrightarrow{PO}+\overrightarrow{OA})\cdot (\overrightarrow{PO}+\overrightarrow{OB})=(\overrightarrow{PO}+\overrightarrow{OA})\cdot (\overrightarrow{PO}-\overrightarrow{OA})=\overrightarrow{PO}\cdot \overrightarrow{PO}-\overrightarrow{OA}\cdot \overrightarrow{OA}=r^2-r^2=0.$$

The result shows that  $\overrightarrow{PA}⟂\overrightarrow{PB}$, i.e. the circumferential angle $APB$ of the half-circle is a right angle.