Are they equal?

This question was proposed by Tony García on Facebook. Here is a solution that includes Mathematica calculations. The Descartes formula for circles touching other three mutually tangent circles is also used.

Solution

These are the values of R₁ (radius of upper red circle) and R₂ (radius of lower red circle) in terms of the side a of the square.

Locus of centroid of equilateral triangles in a parabola

Problem. Find the locus of centroid of equilateral triangles inscribed in the parabola $x^2=4ay$.

To solve this problem we use complex numbers and Mathematica's Eliminate. We find that the locus is another parabola.

 Locus of centroid of equilateral triangles inscribed in a parabola