We start with a triangle with vertices $A$, $B$, and $C$ on a given circumference.
Now we want to find the midpoints of the sides: $M_{AB}$ (midpoint of $AB$), $M_{BC}$ (midpoint of $BC$), and $M_{CA}$ (midpoint of $CA$). Draw lines from vertex $A \rightarrow M_{BC}$, from vertex $B \rightarrow M_{CA}$, and from vertex $C\rightarrow M_{AB}$, extending them to intersect the circumference at points $A'$, $B'$, and $C'$, respectively. Construct a new triangle $\triangle A'B'C'$.
I want to proof the triangle converge to an equilateral triangle. One idea I had is proof that in each iteration the area increase, but I couldn’t prove that yet. Could some one help me with some idea.
I've discovered this property long time ago I did some simulations and it seems area increase in each iteration.
