Let $ABC$ be a right triangle acutangle and let $\overline{ AD}$, with D in $\overline{BC}$, be a height relative to point $A$. Let $Γ_1$ and $Γ_2$ as circumferences circumscribed to triangles $ABD$ and $ACD$, respectively. The circumference $Γ_1$ crosses the $AC$ side at points $ A$ and $P$, while $Γ_2$ crosses the AB side at points $B$ and $Q$. Let $X$ or line intersection point BP with $Γ_2$ so that $P$ is between $B$ and $X$. Likewise, be $Y$ the intersection point of the line $QC$ with $Γ_1$ so that $ Q $ is between $C$ and $Y$. Knowing that $A, X$ and $Y$ are collinear, calculate the smallest possible value to measure the $\angle{BAC}$ angle.
I think triangular inequality would help