examday

/TMUA

Triangle $ABC$ is right-angled at $A$, with $AB=AC=5$.

Three circles have centres $A$, $B$ and $C$ and radii $r$, $2r$ and $\sqrt2\,r$, respectively, where $r>0$.

Every point inside or on the boundary of triangle $ABC$ is required to lie inside or on at least one of the three circles.

What is the least possible value of $r^2$?