Let $P_1: ax^2 + bx +c$ and $P_2: \alpha x^2 + \beta x + \gamma$ be two parabolas that intersect at two points, and let $l$ be the line joining the points of intersection. Between any pair of vertical lines, let $A_1$ be the area bounded by $P_1$ and $l$ and $A_2$ be the area bounded by $P_2$ and $l$. Then $$\frac{A_1}{A_2} = \left| \frac{a}{\alpha} \right|$$
I proved the above statement which should be a high school theorem, but I was not able to find any mentioning of it anywhere. If anyone has seen this statement or a proof of it, please direct me to the link or reference where it can be found.
The area bounded by the parabolas and the line between any pair of vertical lines through $p_1$ and $p_2$:
