Question 145
Let $ABC$ be a triangle with orthocentre $H$. Let $\left(d_{1}\right)$ and $\left(d_{2}\right)$ be two perpendicular lines meeting at $H$. Let $A_{1}$ (respectively $B_{1}, C_{1}$) be the intersection of $\left(d_{1}\right)$ with $(BC)$ (respectively $(CA), (AB)$). Let $A_{2}$ (respectively $B_{2}, C_{2}$) be the intersection of $\left(d_{2}\right)$ with $(BC)$ (respectively $(CA), (AB)$). Let $A_{3}, B_{3}, C_{3}$ be points in the plane such that $A_{1}A_{2}A_{3}, B_{1}B_{2}B_{3}, C_{1}C_{2}C_{3}$ are directly similar triangles.
Show that $A_{3}, B_{3}, C_{3}$ are collinear.
Show that $A_{3}, B_{3}, C_{3}$ are collinear.