Let $H$ be the orthocentre of an equilateral triangle $ABC$ and $J$ be the orthocentre of the triangle $HBC$. $AB=6\sqrt3$. Find the length of $AJ$.
My friend thinks that $A$ and $J$ will align themselves, making the length of the segment $AJ$ will $0$. I don't quite get him and his theory of how they can coincide. Is he right? If no, how can I solve it?
But $A\equiv J$, which says that $AJ=0$.
Indeed, $AH\perp BC$, $AC\perp BH$ and $AB\perp CH$,
which says that $A$ is an ortocenter of $\Delta BHC$, which unique for all triangle.