Let $DEFG$ be the square with $A_1$ as centre such that $D, E$ lie on $BC$, $F$ lies on $CA$, and $G$ lies on $AB$. By considering a homothety with centre $A$, we can map $DEFG$ to a square $D'E'CB$ since $GF // BC$. Let $X$ be the centre of $D'E'CB$. Due to the homothety, $A$, $A_1$, $X$ are collinear. Similarly, let $Y$ and $Z$ be the centres of the squares constructed outside $\triangle ABC$ having $CA$ and $AB$ as a side respectively. It suffices to prove $AX$, $BY$, $CZ$ are concurrent. This follows from Jacobi's theorem since $\angle ZAB = \angle YAC = 45^\circ$, $\angle XBC = \angle ZBA = 45^\circ$ and $\angle YCA = \angle XCB = 45^\circ$.