A moving triangle Three particles A, B, and C are moving in the plane along parallel lines, at constant speeds. (The speeds of the particles are not necessarily the same.) At the start, the area of triangle ABC is $2$. Five seconds later, the area of triangle ABC is $3$. Let $k$ be the area of the triangle in five more seconds. Find the sum of all possible values of $k$. I have never solved a question like this. Though, i tried doing some calculation with the particles respective speeds, but it looked bad(taking three speeds).
 A: One of the ways:
WLOG, particles are moving along axis $OX$.
Then coordinates of them are:
$\qquad A(x_A,y_A)\qquad\rightarrow\qquad A'(x_A+a,y_A)\qquad\rightarrow\qquad A''(x_A+2a,y_A)$;
$\qquad B(x_B,y_B)\qquad\rightarrow\qquad B'(x_B+b,y_B)\qquad\rightarrow\qquad B''(x_B+2b,y_B)$;
$\qquad C(x_C,y_C)\qquad\rightarrow\qquad C'(x_C+c,y_C)\qquad\rightarrow\qquad C''(x_C+2c,y_C)$.
It is comfortable to calculate area of triangle using determinant (see here, formula (16)).
So, (signed) area of initial triangle is
$$\Delta_{ABC}=\begin{array}{|ccc|}x_A & y_A & 1 \\ x_B & y_B & 1 \\ x_C & y_C & 1\end{array}=\pm 2,\tag{1}$$
(signed) area of triangle $A'B'C'$ is
$$\Delta_{A'B'C'}=\begin{array}{|ccc|}x_A+a & y_A & 1 \\ x_B+b & y_B & 1 \\ x_C+c & y_C & 1\end{array}
=\begin{array}{|ccc|}x_A & y_A & 1 \\ x_B & y_B & 1 \\ x_C & y_C & 1\end{array}
+\begin{array}{|ccc|}a & y_A & 1 \\ b & y_B & 1 \\ c & y_C & 1\end{array}
=\Delta_{ABC}+\delta=\pm 3,\tag{2}$$
where we denoted
$$
\delta = \begin{array}{|ccc|}a & y_A & 1 \\ b & y_B & 1 \\ c & y_C & 1\end{array};
$$
and (signed) area of triangle $A''B''C''$ is (similar)
$$\Delta_{A''B''C''}=\begin{array}{|ccc|}x_A+2a & y_A & 1 \\ x_B+2b & y_B & 1 \\ x_C+2c & y_C & 1\end{array}=\Delta_{ABC}+2\delta.\tag{3}$$
Possible cases:


*

*$\Delta_{ABC}=+2,\Delta_{A'B'C'}=+3$ $\;\Rightarrow\; \delta=1$ $\;\Rightarrow\; \Delta_{A''B''C''}=4$;

*$\Delta_{ABC}=+2,\Delta_{A'B'C'}=-3$ $\;\Rightarrow\; \delta=-5$ $\;\Rightarrow\; \Delta_{A''B''C''}=-8$;

*$\Delta_{ABC}=-2,\Delta_{A'B'C'}=+3$ $\;\Rightarrow\; \delta=5$ $\;\Rightarrow\; \Delta_{A''B''C''}=8$;

*$\Delta_{ABC}=-2,\Delta_{A'B'C'}=-3$ $\;\Rightarrow\; \delta=-1$ $\;\Rightarrow\; \Delta_{A''B''C''}=-4$.


So, possible values for (unsigned) $k$ are $4$ and $8$.
