Question 10
Given a triangle $ABC$, let $B^{\prime}$ and $C^{\prime}$ be points on the sides $AB$ and $AC$ such that $BB^{\prime}=CC^{\prime}$. Let $O$ and $O^{\prime}$ be the circumcentres (the centres of the circumscribed circles) of $ABC$ and $AB^{\prime}C^{\prime}$, respectively. Suppose $OO^{\prime}$ intersects lines $AB^{\prime}$ and $AC^{\prime}$ at $B^{\prime\prime}$ and $C^{\prime\prime}$, respectively. If $AB=\frac12 AC$, then
