23. Form the equation of the line passing through two points $A\left(x_{1} ; y_{1}\right)$ and $B\left(x_{2} ; y_{2}\right)$
Solution. Let $C(x ; y)$ be the point of the desired line. Then consider the degenerate triangle $A B C$ (Problem 20). Setting $x_{3}=x$ and $y_{3}=y$, we get:

$$
\boldsymbol{x}_{1} y_{2}+x_{2} y+x y_{1}=y_{1} x_{2}+y_{2} x+y x_{1}, \text{ or }\left(x-x_{1}\right)\left(y_{2}-y_{1}\right)=\left(y-y_{1}\right)\left(x_{2}-x_{1}\right) .
$$