USING MATH TO HELP YOU SOLVE THE NUMBER PUZZLE

You can use math to help you solve puzzles like this!
The technique is illustrated with an example.

Suppose you are given the digits $\,1\,$ through $\,9\,,$ and you want each side of the triangle to sum to $\,23\,.$

If we can figure out the numbers that go in the corners, then it's easy from there.

Let the numbers in the corners be $\,a\,,$ $\,b\,,$ and $\,c\,.$
Let $\,x\,$ represent the sum of the numbers between $\,a\,$ and $\,b\,.$
Let $\,y\,$ represent the sum of the numbers between $\,a\,$ and $\,c\,.$
Let $\,z\,$ represent the sum of the numbers between $\,b\,$ and $\,c\,.$

Since the sum along each side must equal $\,23\,,$ we have $$ (a + x + b) + (a + y + c) + (b + z + c) = 3(23) = 69 $$ Solving this equation for $\,x+y+z\,$ gives
$$ x + y + z = 69 - 2(a + b + c) $$ Since the digits $\,1\,$ through $\,9\,$ must be used to fill all the circles, we also have $$ a + x + b + z + c + y = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45 $$ Solving this equation for $\,x+y+z\,$ gives $$ x + y + z = 45 - (a + b + c) $$ Let $\,S := a + b + c\,.$
Equating the two expressions for $\,x+y+z\,$ gives $$ 45 - S = 69 - 2S $$ Solving for $\,S\,$ yields $\,S = 24\,.$

The only three available numbers that sum to $\,24\,$ are $\,7\,,$ $\,8\,,$ and $\,9\,,$ so these must go in the corners.

Then, the rest is easy!