Ryan Morgan conjecture
The sides of the triangle are evenly split in n parts where n is odd.
The intersection of the (n-1)/2 and (n+1)/2 segments form a hexagon. The ratio of the area of the triangle to that of the hexagon seems to be integral.
Geometry Theory Adds Up for Baltimore Youth, 15 (LA Times)
The ratios observed are :
10, 28, 55, 91, 136, 190, 253, 325, 406,...
Which seems to be every third triangular number
http://oeis.org/A060544