Why is true that $\operatorname{lcm}(\gcd(c_1,c_3),\gcd(c_2,c_3))=\gcd(\operatorname{lcm}(c_1,c_2),c_3)$ ?
LHS is $\displaystyle\frac{\gcd(c_1,c_3)\gcd(c_2,c_3)}{\gcd(c_1,c_2,c_3)}$
RHS is $\displaystyle\gcd\left(\frac{c_1c_2}{\gcd(c_1,c_2)},c_3\right)$
Since $c_1$ and $\frac{c_2}{\gcd(c_1,c_2)}$ are coprime, I can use the distributive property and write RHS as
$$\gcd(c_1,c_3)\gcd\left(\frac{c_2}{\gcd(c_1,c_2)},c_3\right)$$
So if I compare both formula and cancel $\gcd(c_1,c_3)$ from both sides, the following must hold
$$\frac{\gcd(c_2,c_3)}{\gcd(c_1,c_2,c_3)}=\gcd\left(\frac{c_2}{\gcd(c_1,c_2)},c_3\right)$$
It seems, that I obtained a more complicated equality to prove
$\newcommand{\lcm}[0]{\mathrm{lcm}}$The proof of $$ \gcd(\lcm(c_{1}, c_{2}), c_{3}) = \lcm(\gcd(c_{1}, c_{3}), \gcd(c_{2}, c_{3})). $$ is arguably best done using the formula for the gdc and lcm in terms of factorization as product of prime powers. That is, if $p_{k}$ are distinct primes, and $$ c_{i} = \prod_{k} p_{k}^{f_{ik}}, $$ then $$ \gcd(c_{i}, c_{j}) = \prod_{k} p_{k}^{\min(f_{ik}, f_{jk})}, \quad\text{and}\quad \lcm(c_{i}, c_{j}) = \prod_{k} p_{k}^{\max(f_{ik}, f_{jk})}. $$
This way you see immediately that you can reduce to the case when all $c_{i}$ are power of the same prime $p$, that is, $c_{i} = p^{e_{i}}$ for some $e_{i}$.
Now this is easily seen to boil down to the fact that max and min distribute with respect to each other, in particular for this case $$ \min(\max(e_{1}, e_{2}), e_{3}) = \max(\min(e_{1}, e_{3}), \min(e_{2}, e_{3})). $$ This you deal with just distinguishing the $3$ possible cases (noting the symmetry between $e_{1}$ and $e_{2}$) for the relative ordering of the $e_{i}$.