Challenge Level

No the title is not a spelling mistake! This is a beautiful
result involving a parabola and parallels. 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 at *$C$* .

Let $A=(a,a^2)$, $B= (b, b^2 )$, and $C= (c, c^2 )$. 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.