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$$u = \frac{\sqrt{a^2+(r+t)^2}-\sqrt{a^2+(r-t)^2}}{2r}$$ How does one solve for $t$ in terms of all the other variables?

I tried the following SageMath code but failed to find the solution. What is the correct script?

u,r,t,a = var('u','r','t','a') 
g = (sqrt(a^2 + (r+t)^2) - sqrt(a^2 + (r-t)^2))/(2*r) 
ae = (g==u)
view(solve(ae,t))

The output is:

sqrt(a^2 + r^2 + 2rt + t^2) == 2ru + sqrt(a^2 + r^2 - 2rt + t^2)

This is not the desired solution.

$$u = \frac{\sqrt{a^2+(r+t)^2}-\sqrt{a^2+(r-t)^2}}{2r}$$ How does one solve for $t$ in terms of all the other variables?variables using SageMath?

Note: The purpose of this problem is not to find the solution per se, since we can solve it easily by hand, but to solve it completely by the machine algebra in SageMath.

I tried the following SageMath code but failed to find the solution. What is the correct script?

u,r,t,a = var('u','r','t','a') 
g = (sqrt(a^2 + (r+t)^2) - sqrt(a^2 + (r-t)^2))/(2*r) 
ae = (g==u)
view(solve(ae,t))

The output is:

sqrt(a^2 + r^2 + 2rt + t^2) == 2ru + sqrt(a^2 + r^2 - 2rt + t^2)

This However, this is not the desired solution.

$$u = \frac{\sqrt{a^2+(r+t)^2}-\sqrt{a^2+(r-t)^2}}{2r}$$ How does one solve for $t$ in terms of all the other variables using SageMath?

Note: The purpose of this problem is not to find the solution per se, since we can solve it easily by hand, but to solve it completely by the machine algebra in SageMath.SageMath. I am having difficulty coaxing SageMath to do that.

I tried the following SageMath code but failed to find the solution. What is the correct script?

u,r,t,a = var('u','r','t','a') 
g = (sqrt(a^2 + (r+t)^2) - sqrt(a^2 + (r-t)^2))/(2*r) 
ae = (g==u)
view(solve(ae,t))

The output is:

sqrt(a^2 + r^2 + 2rt + t^2) == 2ru + sqrt(a^2 + r^2 - 2rt + t^2)

However, this is not the desired solution.

$$u = \frac{\sqrt{a^2+(r+t)^2}-\sqrt{a^2+(r-t)^2}}{2r}$$ How does one solve for $t$ in terms of all the other variables using SageMath?

Note: The purpose of this problem is not to find the solution per se, since we can solve it easily by hand, but to solve it completely by the machine algebra in SageMath. I am having difficulty coaxing SageMath to do that.

I tried the following SageMath code but failed to find the solution. What is the correct script?

u,r,t,a = var('u','r','t','a') 
g = (sqrt(a^2 + (r+t)^2) - sqrt(a^2 + (r-t)^2))/(2*r) 
ae = (g==u)
view(solve(ae,t))

The output is:

sqrt(a^2 + r^2 + 2rt + t^2) == 2ru + sqrt(a^2 + r^2 - 2rt + t^2)

However, this is not the desired solution.