u/Nomadic_Seth

I've been thinking about a classical result in conic geometry that I think deserves more attention.

Take the parabola x² = 4ay. From any point Q = (h, k) inside the evolute, you can draw exactly three normals to the curve. Each normal meets the parabola at a foot, giving you three points and those three points form a triangle.

The theorem: the centroid of that triangle always lies on the axis of the parabola.

The proof comes down to one beautiful observation. When you substitute Q into the normal equation x + ty = 2at + at³, you get the cubic

at³ + (2a − k)t − h = 0

There is no t² term. By Vieta's formulas, the sum of the roots is zero: t₁ + t₂ + t₃ = 0. Since the x-coordinate of the centroid is (2a/3)(t₁ + t₂ + t₃), it vanishes identically.

What's even nicer: the y-coordinate of the centroid works out to 2(k − 2a)/3 — it depends only on k, the height of Q. The horizontal position h disappears entirely. So if you slide Q left and right at fixed height, the centroid doesn't move at all. That's what the GIF shows.

Here’s the full video:

https://youtu.be/BT8ByfN1SNo?si=Fo8Ii_KQOvWwSzjn