Answer
$a=0.366 L$
Work Step by Step
Due to symmetry, $A_y=B_y$
$
\begin{array}{ll}
+\uparrow \Sigma F_y=0 ; & A_y+B_y-\frac{w(L-a)}{4}-w a-\frac{w(L-a)}{4}=0 \\
& A_y=B_y=\frac{w}{4}(L+a) \\
C+\Sigma M=0 ; &-M-\frac{w a}{2}\left(\frac{a}{4}\right)-\frac{w(L a)}{4}\left(\frac{a}{2}+\frac{L}{6}-\frac{a}{6}\right)+\frac{w}{4}(L+a)\left(\frac{a}{2}\right)
\end{array}
$$
Since $M=0$;
$$
\begin{aligned}
& 3 a^2+(L-a)(L+2 a)-3 a(L+a)=0 \\
& 2 a^2+2 a L-L^2=0 \\
& a=0.366 L
\end{aligned}
$
