PSL4#58

let the vertices of the triangle be labeled a, b, c, and the midpoints d, e, f

 

let the sum of a given leg equal S

 

3S = 3(a+b+c) + (d+e+f)

 

S = (a+b+c) + (d+e+f)/3

 

Note that n=(d+e+f)/3 is an integer

 

n cannot be 1 as the minimum value is 2

 

if n=2, the only possible values of d, e, f are (1,2,3)

 

if n=3, the possible values of d, e, f are (1,2,6), (1, 3, 5), (2, 3, 4)

 

Setting d, e, and f, fixes what you can use for a, b, and c as any different combos will be rotations or reflections.

 

Thus you have 4 possible ways to arrange the triangle.