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 $\overrightarrow{PA} \perp \overrightarrow{PB}$ , i.e. the
circumferential angle$APB$ of the half-circle is a right angle.
"proving Thales' theorem with vectors" is owned by pahio.
![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
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