One of the difficulties is the 0 at the end. If you take it out, then the problem becomes to put the numbers 4, 5, 6, 7, 8, 9 as you say on the triangle, so that the numbers on a line add up to 20. But now, you have a clue: To get a sum of 20 with three numbers, you need either three even numbers, or two odd and an even.

In all you have three odd numbers: 5, 7, 9, and three even numbers: 4, 6, 8. Since 4 + 6 + 8 is 18, not 20, the three even numbers will not be on the same line, so each line will contain two odd numbers and an even number.

Do you see how this helps?

For one thing, you can simplify the problem by dropping the final zeros -- let the numbers be 4 through 9, and try for totals of 20 per edge. The next thing is to note that if you add together the numbers along each side you get 20 + 20 + 20 = 60. But you have added each corner number twice and each side number once, thus

It seems as if you'll need the corner numbers to add up to 21, and the others add up to 18.