$V_A = V_B = \frac{wl}{2}$

B.M at C

$B.M_c = 0$

$\frac{wl}{2} \times (\frac{l}{2}) - H \times h - \frac{wl}{2} \times \frac{l}{4} = 0$

$hH = \frac{wl^2}{4} - \frac{wl^2}{8} = \frac{2wl^2}{8} - \frac{wl^2}{8}$

$\boxed{H = \frac{wl^2}{8h}}$

Consider part (xA)

$B.M_x = \frac{wl}{2} x - \frac{wl^2}{8h}y - wx.\frac{x}{2}$

But we know that $y= \frac{4h}{l^2}x(l-x) = \frac{4h}{l^2}(lx - x^2)$

$\frac{wl}{2} x - \frac{wl^2}{8h} \times \frac{4h}{l^2} (lx - x^2) - \frac{wx^2}{2}$

$\frac{wl}{2}x - \frac{w}{2}(lx - x^2) - \frac{wx^2}{2}$

$\frac{wlx}{2} - \frac{wlx}{2} + \frac{wx^2}{2} - \frac{wx^2}{2} = 0$

Hence prove that BM at every section is 0.