This is from a problem seminar and I need help figuring out the solution.
Four bugs, $A,B,C,D$ are initially placed at the corners of a unit square. From a given initial moment, all four crawl simultaneously at one and the same speed $s>0$, $A$ towards $B$, $B$ towards $C$, $C$ towards $D$, and $D$ towards $A$, with each heading at every instant along the line joining it with its target.
Find a system of differential equations describing the trajectory of $A$.
Do the bugs eventually all meet at the center of the square? If so, how long did it take?
My attempt so far: $\def\grad{\mathbf\nabla}$ $\newcommand{\norm}[1]{\left\Vert #1 \right\Vert}$ Call ${\vec x}_1,{\vec x}_2,{\vec x}_3,{\vec x}_4$ the positions in the plane of $A,B,C,D$ respectively. Then $\grad {\vec x}_1 = \frac{{\vec x}_2-{\vec x}_1}{\norm{{\vec x}_2-{\vec x}_1}}s$ $\grad {\vec x}_2 = \frac{{\vec x}_3-{\vec x}_2}{\norm{{\vec x}_3-{\vec x}_2}}s$
$\grad {\vec x}_3 = \frac{{\vec x}_4-{\vec x}_3}{\norm{{\vec x}_4-{\vec x}_3}}s$
$\grad {\vec x}_4 = \frac{{\vec x}_1-{\vec x}_4}{\norm{{\vec x}_1-{\vec x}_4}}s$
with ${\vec x}_1(0) = (0,0),\ {\vec x}_2(0) = (0,1),\ {\vec x}_3(0) = (1,1),\ {\vec x}_4(0) = (1,0)$.
I have noticed that the sum of the gradients is 0, but I don't know if that is helpful or not. Can someone help me figure this problem out?