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Particular solve: find values of variables which will solve equation for any values of other variables

Hi, this is my first time using Sage, so I hope my questions aren't too trivial.

I have to solve the following problem: find (if they exist) values of real variables x,u,v (with x,v>0) such that the following equality

Maximum{2 a/x, 2 (ub + vc)/(u^2 + v^2), 2 ((u - x)(b - a) + vc)/(v^2 + (u - x)^2)} = (a/x + c/v) + (1/vx)sqrt( a^2 (u^2 + v^2) - 2 ax(bu + cv) + (b^2 + c^2)*x^2)

is verified for every value of the real values a, b, c. It is my understanding that to evaluate the maximum I should use max_symbolic, am I right?

Let me emphasize that I don't need to find values of x,u,v with respect to a,b,c, but fixed values of x,u,v which will verify the equality for any given a,b,c. This means that I need to find x, u, v which will verify all the infinite equations we will get for any different triple of values of a, b, c.

For this reason writing

x,u,v,a,b,c=var('x u v a b c') 

s1=solve([max_symbolic(2 a/x, 2*(u*b + v*c)/(u^2 + v^2), 2 ((u - x)*(b - a) + v*c)/(v^2 + (u - x)^2)) == (a/x + c/v)  +(1/v*x)*sqrt(a^2 (u^2 + v^2) - 2 a*x*(b*u + c*v) + (b^2 + c^2)*x^2)],x,u,v)


will not work. I've thought of solving for all 6 variables, but then an error occurred saying solve() could only use 5 positional arguments..

Can you suggest me any way to proceed? Also, I fear this kind of computation will be very heavy for my cpu.. Any ideas on how to make It lighter?

Thank you very much!