Problem
GEO-B2-M09-P024 The General Euler Line
In an arbitrary triangle choose coordinates \(A(0,0)\), \(B(1,0)\), \(C(u,v)\), \(v\ne 0\). Prove that the circumcenter \(O\), centroid \(G\), and orthocenter \(H\) lie on one line, and \(OG:GH=1:2\).
Use the coordinates \(O\left(\frac12,\frac{u^2+v^2-u}{2v}\right)\), \(H\left(u,\frac{u(1-u)}{v}\right)\), \(G\left(\frac{1+u}{3},\frac v3\right)\).
We have \(O\left(\frac12,\frac{u^2+v^2-u}{2v}\right)\), \(H\left(u,\frac{u(1-u)}{v}\right)\), \(G\left(\frac{1+u}{3},\frac v3\right)\). Check that \(G=\frac{2O+H}{3}\). For the first coordinate: \(\frac{2\cdot\frac12+u}{3}=\frac{1+u}{3}\). For the second coordinate: \(\frac{\frac{u^2+v^2-u}{v}+\frac{u-u^2}{v}}{3}=\frac{v}{3}\). Hence \(G\) lies on \(OH\) and divides segment \(OH\) in the ratio \(OG:GH=1:2\).
The final problem of the module: coordinates prove a general theoretical fact.