[tex]a)A(-2,1),B(-1,2),C(2,-1)[/tex]
[tex]A(x_{A},y_{A}),B(x_{B},y_{B}),C(x_{C},y_{C})[/tex]
[tex]A,B,C \: coliniare \: daca \: :[/tex]
[tex]\begin{vmatrix}
x_{A}& y_{A} & 1\\
x_{B}& y_{B} & 1\\
x_{C} & y_{C} & 1
\end{vmatrix} = 0[/tex]
[tex]\begin{vmatrix}
- 2& 1& 1\\
- 1& 2& 1\\
2& - 1& 1
\end{vmatrix} = 0[/tex]
[tex]-2 \times 2 \times 1+(-1) \times (-1) \times 1+2 \times 1 \times 1-1 \times 2 \times 2-1 \times (-1) \times (-1)-1 \times 1 \times (-1) = 0 [/tex]
[tex] - 4 + 1 + 2 - 4 - 1 + 1 = 0[/tex]
[tex]-5 = 0 \: (F)=>A,B,C \:necoliniare[/tex]
[tex]b)M(1,2),N(-1,4),P(1,2)[/tex]
[tex]M(x_{M},y_{M}),N(x_{N},y_{N}),P(x_{P},y_{P})
[/tex]
[tex]M,N,P \: coliniare \: daca :[/tex]
[tex]\begin{vmatrix}
x_{M}& y_{M} & 1\\
x_{N}& y_{N} & 1\\
x_{P} & y_{P} & 1
\end{vmatrix} = 0[/tex]
[tex]\begin{vmatrix}
1& 2 & 1\\
- 1& 4& 1\\
1 & 2 & 1
\end{vmatrix} = 0[/tex]
[tex]1 \times 4 \times 1+(-1) \times 2 \times 1+1 \times 2 \times 1-1 \times 4 \times 1-1 \times 2 \times 1-1 \times 2 \times (-1) = 0[/tex]
[tex]4 - 2 + 2 - 4 - 2 + 2 = 0[/tex]
[tex]0 = 0 \: (A)=>M,N,P\: coliniare[/tex]
[tex]c)E(2,1),F(-1,3),Q(2,3)[/tex]
[tex]E(x_{E},y_{E}),F(x_{F},y_{F}),Q(x_{Q},y_{Q})[/tex]
[tex]E,F,Q \: coliniare \: daca :[/tex]
[tex]\begin{vmatrix}
x_{E}& y_{E} & 1\\
x_{F}& y_{F} & 1\\
x_{Q} & y_{Q} & 1
\end{vmatrix} = 0[/tex]
[tex]\begin{vmatrix}
2& 1 & 1\\
- 1& 3& 1\\
2 & 3 & 1
\end{vmatrix} = 0[/tex]
[tex]2 \times 3 \times 1+(-1) \times 3 \times 1+2 \times 1 \times 1-1 \times 3 \times 2-1 \times 3 \times 2-1 \times 1 \times (-1) = 0[/tex]
[tex]6 - 3 + 2 - 6 - 6 + 1 = 0[/tex]
[tex] - 6 = 0 \:(F)=>E,F,Q \:necoliniare[/tex]
