Question 84
Let $ABC$ be an isosceles triangle such that $|AB| = |AC|$ and $\angle BAC \lt 60^\circ$. Let $D$ be the point on the segment $\overline{AC}$ such that $\angle DBC = \angle BAC$, $E$ be the intersection of the perpendicular bisector of the segment $\overline{BD}$ and the line passing through $A$ parallel to $BC$, and $F$ be the point on the line $AC$ such that $A$ lies between $C$ and $F$, and $|AF| = 2|AC|$ holds.
a) Prove that the lines $BE$ and $AC$ are parallel.
b) Prove that the line passing through $F$ perpendicular to $AB$ and the line passing through $E$ perpendicular to $AC$ intersect on the line $BD$.
It is allowed to use the claim from a) in b) even if it is not proven. (Italy 2013)
a) Prove that the lines $BE$ and $AC$ are parallel.
b) Prove that the line passing through $F$ perpendicular to $AB$ and the line passing through $E$ perpendicular to $AC$ intersect on the line $BD$.
It is allowed to use the claim from a) in b) even if it is not proven. (Italy 2013)