Food for thought

Take any two points $A$ and $B$ on the parabola $y = x^2$.
Draw the line $OC$ through the origin, parallel to $AB$, cutting the parabola again at $C$.
Let $A$ have coordinates $(a,a^2)$, let $B$ have coordinates $(b,b^2)$ and let $C$ have coordinates $(c,c^2)$.
Prove that $a+b = c$.
Imagine drawing another parallel line $DE$, where $D$ and $E$ are two other points on the parabola. Extend the ideas of the previous result to prove that the midpoints of each of the three parallel lines lie on a straight line.