Simplify[Solve[{f==Sqrt[y^2+(x-Xl)^2] + Sqrt[x^2+y^2],g==Sqrt[y^2+(x-Xr)^2] + Sqrt[x^2+y^2]}, {x,y}]]

{{x->(((f^2) g-g (Xl^2)+f (-(g^2)+(Xr^2)))/(-2 g Xl+2 f Xr)),y->-((Sqrt[-(((g^2) ((f^2)-(Xl^2)) ((f^2)-2 f g+(g^2)-(((Xl-Xr))^2)) ((g^2)-(Xr^2)))/(((g Xl-f Xr))^2))])/(2 g))},{x->(((f^2) g-g (Xl^2)+f (-(g^2)+(Xr^2)))/(-2 g Xl+2 f Xr)),y->((Sqrt[-(((g^2) ((f^2)-(Xl^2)) ((f^2)-2 f g+(g^2)-(((Xl-Xr))^2)) ((g^2)-(Xr^2)))/(((g Xl-f Xr))^2))])/(2 g))}}

position[f_,g_]:={(((f^2) g-g (Xl^2)+f (-(g^2)+(Xr^2)))/(-2 g Xl+2 f Xr)),((Sqrt[-(((g^2) ((f^2)-(Xl^2)) ((f^2)-2 f g+(g^2)-(((Xl-Xr))^2)) ((g^2)-(Xr^2)))/(((g Xl-f Xr))^2))])/(2 g))}

distance[x_,y_]:={Sqrt[y^2+(x-Xl)^2] + Sqrt[x^2+y^2],Sqrt[y^2+(x-Xr)^2] + Sqrt[x^2+y^2]}

distance[0.3,7.4]

position[14.919400087018944`,14.839112958173462`]

Simplify[Solve[{f==Sqrt[y^2+(x-Xl)^2] + Sqrt[x^2+y^2],g==Sqrt[y^2+(x-Xr)^2] + Sqrt[x^2+y^2],a==Sqrt[y^2+(x-Xl)^2],b==Sqrt[y^2+(x-Xr)^2],c==Sqrt[x^2+y^2]}, {x,y,a,b,c}]]

CForm[((Sqrt[-(((g^2) ((f^2)-(Xl^2)) ((f^2)-2 f g+(g^2)-(((Xl-Xr))^2)) ((g^2)-(Xr^2)))/(((g Xl-f Xr))^2))])/(2 g))]