Parabella
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.
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