I am looking for pre-calculus proof for the following scenario:
I have a parabola $F(x)$ and a line that goes through $2$ points on given parabola $G(x)$.
If I trace half the horizontal length of the line and from there create a vertical line going through the parabola and the line, the vertical line will hit the parabola exactly in $F'(x)=G'(x)$.
But why? I've seen that this works but cant put it into mathematical proof.
My scenario on desmos
The red lines are the half the horizontal distance of the line they're standing on
Best Answer
Let $F(x)=ax^2+bx+c$ and $G(x)=kx+m$. Their intersection points are given by,
$$ax^2+(b-k)x +c -m =0$$
Then, the $x$-coordinate of the vertical intersection is
$$x_m = \frac{x_1+x_2}{2}= -\frac {b-k}{2a}$$
Thus,
$$F'(x_m)= 2a x_m + b = k =G'(x_m)$$