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Place the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 in the circles
so that all three sides of the triangle have the same sum.

The numbers 1-9 add up to 45.
The sum of all three sides = 45 + (3 corners),
since each corner contributes to two sides.
Sum of all three sides = 3 * one side.
(3 corners) = sum of all three sides - 45.
Since both terms on the right side are divisible by 3,
the sum of the three corners is divisible by 3 too.
That limits the choices for the corners.
The sum of each side then will be (45 + (3 corners))/3.

From there, exhaustive search reveals the solutions below.


Each solution has a complement, where each number, k, is replaced by 10-k.
In the first two rows, the complements are one above the other,
In the bottom row, they are all self-complementary, with a reflection about the vertical axis.

The last pair are the only ones where, for a given choice for the corners, there is only one solution. The others are all in pairs.

Solutions

Side = 17
   Side = 19
   Side = 19
2
5   4
9     8
1  6 7  3
 2
6   5
8     7
1  4 9  3
 4
5   2
9     6
1  3 8  7
 4
6   3
8     5
1  2 9  7
 3
5   1
9     8
2  4 6  7
 3
6   4
8     5
2  1 9  7
Side = 23
 Side = 21
 Side = 21
8
5   6
1     2
9  4 3  7
 8
4   5
2     3
9  6 1  7
 6
5   8
1     4
9  7 2  3
 6
4   7
2     5
9  8 1  3
 7
5   9
1     2
8  6 4  3
 7
4   6
2     5
8  9 1  3
Side = 20
 Side = 20
 Side = 20
 Side = 20
5
9   1
2     8
4  3 7  6
 5
3   7
8     2
4  1 9  6
 5
4   6
9     1
2  3 7  8
 5
6   4
7     3
2  1 9  8
 5
6   4
8     2
1  3 7  9
 5
4   6
8     2
3  1 9  7