pick point on parabola so 2 conditions are true

pick point on parabola so 2 conditions are true

You have a parabola y=ax^2+bx+c. We know a,b & c. On this parabola you
have to pick a point A where the following conditions are true:
1) if you draw a tangent line in this point to the parabola, then the
middle point of the following 3 points has to be the midpoint of the other
two points (same distance):
-the point where the parabola touches the tangent line (point A)
-the intersection of the tangent line with the horizontal line drawn
through the vertex: y=(4ac-b^2)/4a.
-the intersection of the tangent line with the axis of symmetry = vertical
line through the vertex: x=-b/2a.
2) Normally, then also for the following 3 points the middle point is on
the same distance to the first and to the third point (all be it on a
different distance as Condition 1, depending on the shape of the
parabola).
-the intersection point of the horizontal line through point A and the
axis of symmetry
-the vertex
-the intersection point of the tangent line to the parabola in point A,
with the axis of symmetry
I know that the slope of the tangent line in a certain Ax will be 2a*Ax+b,
so you can calculate the tangent line through A to be equal to
y=(2aAx+b)(x-Ax)+Ay, but I'm not sure how to move from there to pick point
A so all conditions are still true.
All help is truly appreciated. These things can be pretty frustrating if
you're not mathematically inclined..
In this image, pick point A such that |AF| = |FE| and |ED| =|DB|. I can do
it geometrically, but apparently not algebraically.