Find the distance between $B$ and $A$, call this $ {BA}$.
Find the distance between $C$ and $B$ (it's $ {BA}-d$), call this $ {CB}$.
Let the coordinates of $C$ be $(c_1,c_1)$.
Now use similar triangles:
$ {\color{darkblue}{y_1-y_2}\over {BA}}= {\color{darkgreen}{c_2-y_2}\over {CB}} $
and
$ {\color{maroon}{x_1-x_2}\over {BA}}= {\color{orange}{c_1-x_2} \over {CB}}. $
Solve the above for $c_1$ and $c_2$.
Triangles $\triangle ABa$ and $\triangle CBc$ are similar. That is, they have the same angles. Thus, corresponding ratios of side lengths are equal: $\def\overline{} {\overline{BC}\over\overline{ Bc}}={\overline{BA}\over\overline{ Ba}},\quad {\overline{BC}\over\overline{ Cc}}={\overline{BA}\over\overline{ Aa}},\quad {\overline{Bc}\over \overline{Cc}}={\overline{Ba}\over\overline{ Aa}}. $
This is the same as saying the slope of a line can be computed using two arbitrary points on the line.