Problem
GEO-B2-M09-P021 A Vector Formula for the Orthocenter
In triangle \(A(0,0)\), \(B(1,0)\), \(C(u,v)\), \(v\ne 0\), let \(O\) be the circumcenter and \(H\) the orthocenter. Prove that \(\overrightarrow{OH}=\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}\).
Find \(O\) and use the formula \(H\left(u,\frac{u(1-u)}{v}\right)\).
The circumcenter has coordinates \(O\left(\frac12,\frac{u^2+v^2-u}{2v}\right)\). The orthocenter is \(H\left(u,\frac{u(1-u)}{v}\right)\). The vector equality \(\overrightarrow{OH}=\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}\) is equivalent to \(H=A+B+C-2O\). The right-hand side equals \((1+u,v)-\left(1,\frac{u^2+v^2-u}{v}\right)=\left(u,\frac{u-u^2}{v}\right)=H\). The equality is proved.
This is a strong link between coordinates and vectors, useful before the general Euler line.