Problem
GEO-B3-M06-P018 Euler Line Parallel to a Side
In triangle \(ABC\), prove that the Euler line is parallel to \(BC\) if and only if \(\tan B\tan C=3\).
C. Hint 1. Project the condition onto the altitude from \(A\).
D. Hint 2. Use \(AH=2R\cos A\) and \(AA_h=2R\sin B\sin C\).
Let \(A_h\) be the foot of the altitude from \(A\). If the Euler line is parallel to \(BC\), projecting onto \(AA_h\) and using the centroid ratio on the median gives \(AH:AA_h=2:3\).
We have \(AH=2R\cos A\), and \(AA_h=c\sin B=2R\sin C\sin B\). Hence \[ \frac{\cos A}{\sin B\sin C}=\frac23. \] Substituting \(\cos A=\sin B\sin C-\cos B\cos C\), we obtain \(\sin B\sin C=3\cos B\cos C\), i.e. \(\tan B\tan C=3\). Reversing the steps proves the converse.
A strong problem combining classical geometry with trigonometry.