Ellipse 3-partition: same area and perimeter

Let $V_1 V_2$ be the major axis of our ellipse, with area $\Delta$. Take a point $P$ between $V_1$ and $V_2$ such that $PV_1=x$, and two points $Q_1,Q_2$ such that $Q_1 Q_2$ is perpendicular to $V_1 V_2$ and the elliptic sector $E_1$ delimited by the rays $PQ_1,PQ_2$ has area $\Delta/3$. Let $E_2$ be the elliptic sector delimited by the rays $PQ_1,PV_2$ and $p_j(x)$ the perimeter of $E_j$. The function

$$ f(x) = p_1(x) - p_2(x) $$

is clearly continous, so, if we find two points $x_0,y_0$ such that $f(x_0)f(y_0)<0$, we are sure that $f$ has at least a zero $z$ and we have done, since:

$$ \Delta(E_2) = \frac{\Delta-\Delta(E_1)}{2} = \Delta(E_1). $$