On wikipedia in German, we find relations about two angles inscribed on parable and on hyperbole. The 4 points of the parabola $y = ax^2 + bx + c $ has the following property: $$ \frac{(y_4-y_1)}{(x_4-x_1)}-\frac{(y_4-y_2)}{(x_4-x_2)}=\frac{(y_3-y_1)}{(x_3-x_1)}-\frac{(y_3-y_2)}{(x_3-x_2)} $$ http://de.wikipedia.org/wiki/Parabel_%28Mathematik%29#Peripheriewinkelsatz_f.C3.BCr_Parabeln
The 4 points of hyperbole $ y = \frac {a} {x-b} + c$ have the following property (slightly modified) : $$ \frac{(y_4-y_1)}{(x_4-x_1)}/\frac{(y_4-y_2)}{(x_4-x_2)}=\frac{(y_3-y_1)}{(x_3-x_1)}/\frac{(y_3-y_2)}{(x_3-x_2)} $$ http://de.wikipedia.org/wiki/Hyperbel_%28Mathematik%29#Peripheriewinkelsatz_f.C3.BCr_Hyperbeln
But I can not find a similar formula (simple) for the ellipse. Does it exist? I guess we obtain the formula by the cross-ratio?
Converting comment to answer...
I would expect cross-ratio-based relations to be coordinate-independent in a way that these aren't.
These relations may have arisen from simply observing that slopes of chords on $y=x^2$ and $y=1/x$ reduce nicely: Writing $m_{ij}$ for the slope between $(x_i,y_i)$ and $(x_j,y_j)$, we have $$y = x^2 \;\to\; m_{ij} = x_i + x_j\qquad\qquad y = \frac{1}{x}\;\to\;m_{ij}=-\frac{1}{x_ix_j}$$ and then it's pretty easy to get $m_{13}$, $m_{14}$, $m_{23}$, $m_{24}$ to cancel (by addition and subtraction in the first case, and by multiplication and division in the second case).
The fact that $y=x^2$ and $y=1/x$ happen to represent conics could just be a coincidence.