domingo, 30 de diciembre de 2012

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.

lunes, 3 de diciembre de 2012

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