An alternative way to get a contradiction would be:

\(\displaystyle ab = bc \implies a = c , (b \neq 0)\)

\(\displaystyle bc = ac \implies b = a , (c \neq 0)\)

\(\displaystyle a = b = c\)

\(\displaystyle ab = a^2 = 4 \implies a = 2, -2\)

\(\displaystyle \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{3}{a} = \frac{1}{2} \implies a = 6\)