a/b=b/c=c/a = k say,

Then a = bk, b = ck and c = ak

Or a = bk = (ck)k = (ak)kk. So a= ak^3. So k = 1.

Therefore a=bk =b = ck = c

a =b=c.

Let’s assume that the value of the three quotients is the constant k, so that:

a/b=b/c=c/a=k

a/b=k, b*k=a (1)

b/c=k, c*k=b (2)

c/a=k, a*k=c (3)

If we are multiplying (1),(2),(3) then the result will be:

b*k* c*k* a*k=a*b*c

a*b*c*k^3= a*b*c (4)

Because a,b,c different from 0, then their product, a*b*c, is also not cancelling.

Because of this, we’ll divide the relation (4), by the product a*b*c and we'll get:

k^3=1, that means that k=1, so a/b=b/c=c/a=k=1

b*k=a

We’ll substitute k=1 and the relation (1) will become: b*1=a , so b=a

c*k=b

We’ll substitute k=1 and the relation (2) will become: c*1=b , so b=c, but b=a so, from transitivity relation results that c=a.

In the end we'll get:

**a=b=c**