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I found this recently and haven't had much time to study but I have found that the values all have a corresponding OEIS sequence. How might this help in disproving the original statement?

Background and motivation A resolution one way or another to the original question could help to resolve a couple of (presumably not so important) but pesky open problems in number theory. I am preparing a website that I will link to eventually that will give the full background. However, it is too lengthy for this post and in accordance with advice from META MO I will omit it to keep this as brief as possible. Additionally I am not a research level mathematician so please forgive any unintended ignorance when responding to comments.

Background and motivation A resolution one way or another to the original question could help to resolve a couple of (presumably not so important) but pesky open problems in number theory. I am preparing a website that I will link to eventually that will give the full background. However, it is too lengthy for this post and in accordance with advice from META MO I will omit it to keep this as brief as possible. Additionally I am not a research level mathematician so please forgive any unintended ignorance when responding to comments.

Important update

Going back and looking at the background of where this comes from. I found that what I am asking for is equivalent to this,

$$\frac{\left(c_{1}-x_{1}\right)\left(c_{1}+x_{1}\right)}{c_{1}^2}+\frac{\left(c_{2}-x_{2}\right)\left(c_{2}+x_{2}\right)}{c_{2}^2}=\frac{\left(c_{3}-x_{3}\right)\left(c_{3}+x_{3}\right)}{c_{3}^2}$$

Where $c_n$ is the associated hypotenuse of the primitive triple and $x_n$ is an integer solution to the circle,

$$x^2+y^2=2c^2$$

I wanted to mention this connection as it is related to the solution set mentioned by Constantin-Nicolae Beli.

Edit for bounty Joe Silverman and Constantin-Nicolae Beli have already given some good insight into the problem, I am putting a bounty on this with the hope that it will get more attention. I don't have much reputation so doing anything, even just commenting would go a long way for me. Looking at the problem as it stands I see one main problem and two subproblems that may help answer the main problem.

1. Prove the original statement or find a counter example.

Subproblems

1. Does there exist 3 primitive triples such that $\frac{1}{c_1^2}+\frac{1}{c_2^2} = \frac{1}{c_3^2}$?
2. Is the solution set mentioned by Constantin-Nicolae Beli the only solutions to $\frac{a_1b_1}{c_1}+\frac{a_2b_2}{c_2} = \frac{a_3b_3}{c_3}$ and would that also be the same solution set for when the hypotenuse is squared?

Edit for bounty Joe Silverman and Constantin-Nicolae Beli have already given some good insight into the problem, I am putting a bounty on this with the hope that it will get more attention. I don't have much reputation so doing anything, even just commenting would go a long way for me. Looking at the problem as it stands I see one main problem and a subproblem that may help answer the main problem.

Prove the original statement or find a counter example.

Subproblem

Is the solution set mentioned by Constantin-Nicolae Beli the only solutions to $\frac{a_1b_1}{c_1}+\frac{a_2b_2}{c_2} = \frac{a_3b_3}{c_3}$ and would that also be the same solution set for when the hypotenuse is squared?

1. Prove the original statement or find a counter example.

1. Does there exist 3 primitive triples such that $\frac{1}{c_1^2}+\frac{1}{c_2^2} = \frac{1}{c_3^2}$?
2. Is the solution set mentioned by Constantin-Nicolae Beli the only solutions to $\frac{a_1b_1}{c_1}+\frac{a_2b_2}{c_2} = \frac{a_3b_3}{c_3}$ and would that also be the same solution set for when the hypotenuse is squared?

1. Prove the original statement or find a counter example.

1. Does there exist 3 primitive triples such that $\frac{1}{c_1^2}+\frac{1}{c_2^2} = \frac{1}{c_3^2}$?
2. Is the solution set mentioned by Constantin-Nicolae Beli the only solutions to $\frac{a_1b_1}{c_1}+\frac{a_2b_2}{c_2} = \frac{a_3b_3}{c_3}$ and would that also be the same solution set for when the hypotenuse is squared?