I am going through one of the the scan conversion techniques and it mentions the implicit equation of line as follows:
$$F(x,y) = ax + by + c = 0 .$$
The text also mentions that $F(x,y)=0$ if the point is on the line without any proof. I am looking for a proof.
In learning coordinate geometry, we usually start with plugging some $x_{i}$ into $ax+by+c=0$ to find an unknown $y_{i}$ with known $a, b, c$. Then we observe that these ordered-pairs $(x_{i},y_{i})$ are always collinear in Cartesian plane. Here, the process are reversed. A point $(x',y')$ satisfying $F(x',y')=0$ where $F(x,y)=ax+by+c$. Usual text seldom provides a proof. If so, we need to define what is a straight line on Cartesian plane. Teacher may use dynamic software to illustrate the idea by varying a point on the graph and checking the relation.