Question 21
Let $A B C$ be an acute, scalene triangle with orthocentre $H$. Let $\ell_{a}$ be the line through the reflection of $B$ with respect to $C H$ and the reflection of $C$ with respect to $B H$. Lines $\ell_{b}$ and $\ell_{c}$ are defined similarly. Suppose lines $\ell_{a}, \ell_{b}$, and $\ell_{c}$ determine a triangle $\mathcal{T}$.
Prove that the orthocentre of $\mathcal{T}$, the circumcentre of $\mathcal{T}$ and $H$ are collinear.
Prove that the orthocentre of $\mathcal{T}$, the circumcentre of $\mathcal{T}$ and $H$ are collinear.