Question

If
x/y=y/z=z/x
show that x=y=z​

Answer 1

Explanation:

x/y = y/z = z/x

x = y²/z = zy/x and y = z²/x

x = y²/z

putting this into the 3rd part of the original equation :

z/x = z/ y²/z = z²/y²

now using the y identity of the second part there :

y = z²/x

z²/y² = z² / z⁴/x² = x²/z² (= z/x per previous equation)

so,

x²/z² = z/x

x³/z³ = 1

x/z = 1

x = z

then

y/z = y/x = z/x (= 1)

and therefore,

y/x = z/x meaning

y = z (and therefore = x)

so,

x = y = z