I don't know how to draw diagram here, so solution is without any diagram assume we have chosen $8$ vertices satisfying the conditions of the problem. Let the height of each small triangle be equal to $1$ and denote by $a_i, b_i, c_i$ the distance of the $i$ th point from the three sides of the big triangle. For any $i = 1, 2, . . . , 8$ we then have $a_i, b_i, c_i\geq0$ and $a_i +b_i +c_i = 10$. Thus, $(a_1 +a_2 +.....+a_8)+(b_1 +b_2 +.....+b_8)+(c_1 +c_2 +.....+c_8) = 80$. On the other hand, each of the sums in the brackets is not less than $0+1+.....+7 = 28$, but $3.28 = 84 > 80,$ a contradiction.