Equality in its simplest form

Let as an example there be two expressions which are equal having 5 symbolic variables x,y,g,f,c. Now we have two expressions obtained from two given conditions. expression1 T1= ((f^2 + g^2 - c)^(1/2) + (a^2)^(1/2))^2 - g^2 - f^2 and expression T2=((f^2 + g^2 - c)^(1/2) + 2*(a^2)^(1/2))^2 - f^2 - (2*a + g)^2; Now these two expressions are related that is T1 is related to T2 for e.g. T1=T2 or T1=T2^2 or T1^2-T1*T2+T2^2=0. Now whatever may be the relation my objective is to express the resulting polynomial in x,y,g,f,c to be as simple as possible and simple looking. There should be always x and y (a 2 dimensional curve in x and y) and to eliminate as much as possible other symbolic variables. Also there should be no fractional powers and as less terms as possible. as an example if T1=T2 the resulting expression can be simplified to 9*a^2 + 24*a*g + 12*g^2 - 4*f^2. This is the simplest form with c eliminated and least terms with no fractional powers. The solution should be for any general term T1=f1(x,y,g,f,c) and T2=f2(x,y,g,f,c)