Proof:

Step 1:
Firstly, let us prove that a×b=b×c=c×a by showing that a×b=b×c.

Step 2:
Given that a b c=0, we can write the following equation:

a×b+c×a+b×c=0

Step 3:
Rearranging the equation, we get:

a×b=-(c×a+b×c)

Step 4:
Taking the cross product of both sides with b, we get:

a×b×b=-(c×a+b×c)×b

Step 5:
Simplifying the left side using the identity a×b×b=0, we get:

0=-(c×a+b×c)×b

Step 6:
Expanding the right side using the distributive property of cross product, we get:

0=-(c×a)×b-(b×c)×b

Step 7:
Using the identity a×b=-b×a and the distributive property of cross product, we can simplify the right side as follows:

0=c×(b×a)-b×(c×b)

Step 8:
Using the identity a×(b×c)=(a·c)b-(a·b)c and the distributive property of cross product, we can further simplify the right side as follows:

0=c×(a×b)-b×(c×b)=(c·b)a-(b·c)b

Step 9:
Since a b c=0, we have a·b+c·a+b·c=0. Rearranging this equation, we get:

a·b=-(c·a+b·c)

Step 10:
Substituting this equation into the previous equation, we get:

0=(c·b)a+(a·b)b

Step 11:
Using the identity a×b=b×a and the distributive property of cross product, we can simplify the right side as follows:

0=(c·b)a+b×(a·b)

Step 12:
Again using the identity a×(b×c)=(a·c)b-(a·b)c and the distributive property of cross product, we can further simplify the right side as follows:

0=(c·b)a-(b·a)c=c×a-b×c

Step 13:
Therefore, we have shown that a×b=b×c=c×a.