proving Thales' theorem with vectorsProvingThalesTheoremWithVectors2008-02-11 11:26:562008-02-11 11:55:52ProofThales' theorem0% this is the default PlanetMath preamble. as your knowledge
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\rput[a](0,-0.3){$O$}
\rput[a](-2.2,-0.1){$B$}
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\rput[a](1.33,1.8){$P$}
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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.