Prove the following statement

Question

Prove that the point $P=(x',y')$ will be inside the acute or obtuse angle made by $a_1x+b_1y+c_1= 0$ and $a_2x+b_2y+c_2= 0$, if $$(a_1x'+b_1y'+c_1)(a_2x'+b_2y'+c_2)(a_1a_2+b_1b_2)\quad <\text{ or }>\quad0$$

Answer 1

The question is equivalent to: $$\text{The point } P(x',y') \text{ does not lie on the lines }a_1x+b_1y+c_1=0 \text{ and } a_2x+b_2y+c_2=0 \\ \text{ if } (a_1x'+b_1y'+c_1)(a_2x'+b_2y'+c_2)(a_1a_2+b_1b_2)\neq0$$ If the antecedent is true, then $(a_1x'+b_1y'+c_1),(a_2x'+b_2y'+c_2),(a_1a_2+b_1b_2)$ are all non-zero.
$(a_1x'+b_1y'+c_1),(a_2x'+b_2y'+c_2)\neq0$ means that $P(x',y')$ does not satisfy any of the equations of the straight lines so $P$ does not lie on any of the lines.