Question 29
In triangle $ABC$, let $J$ be the centre of the excircle that touches side $BC$ at $A_{1}$ and the extensions of sides $AC$ and $AB$ at $B_{1}$ and $C_{1}$, respectively. The line $A_{1} B_{1}$ meets the line $AB$ perpendicularly at $D$. Let $E$ be the projection of $C_{1}$ onto the line $DJ$. Determine the sizes of the angles $\angle B E A_{1}$ and $\angle A E B_{1}$.