Computational Solution

I only wanted to do the math if I had to, so I made a computer program to solve this problem written in C (view it on GitHub here):
How It Works
Each section is identified and placed in an array based on their letter's alphabetical index. The program then uses a brute-force approach. It works how you would expect, it cycles through every possible permutation and outputs a solution if the sums of each set are equal. You can view the output here.
Example solution
As seen in this diagram a solution is given by the following array in the alphabetical form:
[A, B, C, D, E, F, G, H, I, J, K, L, M, N, O]
= [10, 9, 11, 13, 6, 8, 5, 1, 3, 7, 4, 2, 12, 14, 15]
First Eclipse Sum
A + B + C + D + E + F + G + H = 10 + 9 + 11 + 13 + 6 + 8 + 5 + 1 = 63
Second Eclipse Sum
I + J + M + B + C + E + H + N = 3 + 7 + 12 + 9 + 11 + 6 + 1 + 14 = 63
Third Eclipse Sum
K + L + J + M + C + E + F + D = 4 + 2 + 7 + 12 + 11 + 6 + 8 + 13 = 63
Fourth Eclipse Sum
L + O + M + E + N + H + G + F = 2 + 15 + 12 + 6 + 14 + 1 + 5 + 8 = 63
Other Solutions
I ended up finding many solutions, here are a few in the same alphabetical form:
[A, B, C, D, E, F, G, H, I, J, K, L, M, N, O]
Example 1:
[11, 9, 10, 13, 8, 7, 4, 1, 3, 6, 5, 2, 12, 14, 15]
Example 2:
[9, 11, 10, 13, 7, 8, 4, 1, 3, 5, 6, 2, 12, 14, 15]
Example 3:
[11, 10, 9, 13, 6, 8, 4, 1, 3, 7, 5, 2, 12, 14, 15]
Conclusion
There are many solutions based on my code. If you want to see all of them just compile and run the code linked at the top of this post. Or view the linked youtube video showing a small amount of the many there are. So it is possible and there are many solutions, not just one - given that my code is correct.