An Intersection on a Fixed Line

In the plane of the ellipse $E:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$), let $P(x_0,y_0)$ be a point other than the origin. Through $P$, draw two arbitrary secant lines $AB$ and $CD$ with $A,B,C,D$ all on $E$. Prove that the intersection of the lines $AC$ and $BD$ lies on a fixed line.