Normally the equation of a line is written on standard form, slope-intercept form or point-slope form. However, one can equivalently write out the equation of a line on the form $(a_2 - b_1)(y - b_1) = (b_2 - b_1)(x - a_1)$. This form is apparently useful when trying to define all constructible numbers.
Does anyone know an intuitive proof that the equation of a line can be written on this form (i.e. a proof that is not built on 'reverse engineering' the given equation)?