We can find the coordinates of the corner that lies on the two adjacent sides. Using this we can find all other corners because we are given the coordinates of the center too. Now I'm too lazy to actually do this math but somebody else can do that.

@above not an answer. Figuring out where that corner is the whole point. Also "construct" means "use a compass and straightedge".

Assuming we don't have coordinates, call the center $A$, and the two points $B, C$. Construct the circle $\Gamma$ with diameter $\overline{BC}$ and its perpendicular bisector $\overline{EF}$ where $E, F$ are distinct points on the circle. Let $X, Y$ be the intersections of $\overline{EA}, \overline{FA}$ with $\Gamma$ respectively. This gives $X$ and $Y$ as the two possible corners of the square. The other $3$ corners are simple once we have this

Center $A$ and two points $B$ and $C$. Midpoint $D$ of $BC$. Line $l\ =\ AD$. Trough $B\ :\ b//l$, through $C\ :\ c//l$. Trough $A\ :\ t \bot l, t \cap c=E$ Circle, midpoint $A$, radius $AE$, cuts $l$ in $F$ and $G$. Trough $F\ :\ f//t$, trough $G\ :\ g//t$.