[tex]\displaystyle Se~foloseste~identitatea~lui~Euler: \\ \\ Pentru~orice~punct~M~din~plan~are~loc~relatia \\ \\ \overrightarrow{MA} \cdot \overrightarrow{BC}+\overrightarrow{MB} \cdot \overrightarrow{CA}+ \overrightarrow{MC} \cdot \overrightarrow{AB}=0. ~(*)\\ \\ Fie~H~intersectia~inaltimilor~din~A~si~B.~Atunci~\overrightarrow{HA} \cdot \overrightarrow{BC}=0 \\ \\ si~\overrightarrow{HB} \cdot \overrightarrow{CA}=0. \\ \\ Vrem~sa~demonstram~ca~CH \perp AB. \\ \\ [/tex]
[tex]\displaystyle Tinand~cont~de~relatia~(*)~pentru~M=H,~si~de~faptul~ca \\ \\ \overrightarrow{HA} \cdot \overrightarrow{BC}= \overrightarrow{HB} \cdot \overrightarrow{CA}=0,~va~rezulta~ca~\overrightarrow{HC} \cdot \overrightarrow{AB}=0 \Leftrightarrow HC \perp AB, \\ \\ q.e.d. [/tex]
