If y/(x - z) = (y + x)/z = x/y then find x : y : z ?

Let's solve this step by step:

Given that y/(x - z) = (y + x)/z = x/y

Let's assume y/(x - z) = (y + x)/z = x/y = k (where k is a constant)

From y/(x - z) = k, we get y = k(x - z) ---- (1)

From (y + x)/z = k, we get y + x = kz ---- (2)

From x/y = k, we get x = ky ---- (3)

Now, we have three equations and three variables. We can solve these equations to find the values of x, y, and z in terms of k.

From equation (1), we substitute y from equation (3) into equation (1), we get ky = k(x - z)

This simplifies to x = y + z ---- (4)

Substitute y from equation (3) into equation (4), we get ky = ky + z

This simplifies to z = 0

Substitute z = 0 into equation (2), we get y + x = 0

This simplifies to y = -x

Substitute y = -x into equation (3), we get x = -kx

This simplifies to x = 0

Substitute x = 0 into equation (3), we get y = 0

So, the solution to the given equation is x : y : z = 0 : 0 : 0.