I do not think there is any reason why anyone want to do this, but I will toss it out there. I had a class project to find the solution to the shortest path joining 3 points problem. I thought a subdividing triangle would make sense, but it just find it’s centroid (average the 3 points is the same). After searching close to 50 pages i finial found it to be Fermat point.

This app was created using Processing , which is a low level graphic software that uses Java. You can drag any of the points.

Just in case you happen to know the simple algorithm for creating a subdividing it is:

pt D = A(A,B); // taking average of 2 points (pt A(pt A, pt B) {return P((A.x+B.x)/2,(A.y+B.y)/2);} )

pt E = A(C,B);

pt F = A(A,C);

while((d(D,E) > 1) && (d(D,F) > 1) && (d(E,F) > 1)){

G = A(D,E); // taking average of 2 points (pt A(pt A, pt B) {return P((A.x+B.x)/2,(A.y+B.y)/2);} )

H = A(D,F);

I = A(E,F);

D=G; // store previous number

E=H;

F=I;

}