Question

how would I write "the equation of a line through C that is parallel to AB

Answer 1

you would use the point-slope form of the equation of a line
\[ y-y_0 = m(x-x_0) \]

see http://www.khanacademy.org/math/algebra/linear-equations-and-inequalitie/equation-of-a-line/v/equation-of-a-line-1
for an example