The sides of the triangle are evenly split in n parts where n is odd.[br]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.

The ratios observed are :[br]10, 28, 55, 91, 136, 190, 253, 325, 406,...[br]Which seems to be every third triangular number[br]http://oeis.org/A060544