I want to ask how do I find the locus of this, and what are the techniques to find the locus. The word is very confusing to me, and I searched for it, it said that it is 'In geometry, a locus is a set of all points, whose location satisfies or is determined by one or more specified conditions'. I really don't know how do I find the 'condition'.
Thank you for your help and reply.
Let $\bar{X} = (x ,y)$, $\bar{b} = (b_x, b_y)$ and $\bar{c} = (c_x, c_y)$. Then
$$x=sb_x+tc_x$$ $$y=sb_y+tc_y$$
Solve the linear system for $t$ and $s$,
$$t= \frac{xb_y-yb_x}{c_xb_y-c_yb_x}$$ $$s= \frac{xc_y-yc_x}{b_xc_y-b_yc_x}$$
Use the given $s+t=1$ to obtain
$$\left(\frac{b_y}{c_xb_y-c_yb_x}+ \frac{c_y}{b_xc_y-b_yc_x}\right)x -\left(\frac{b_x}{c_xb_y-c_yb_x}+ \frac{c_x}{b_xc_y-b_yc_x}\right)y=1$$
which shows explicitly that the locus of $\bar{X}$ is a line in two-dimensional space.
Note that, in the general vector form,
$$\bar{X}(s)= s\bar{b}+t\bar{c}=s(\bar{b}-\bar{c}) +\bar{c}$$
which represents a line that passes through the point $\bar{c}$ and is parallel to the direction $\bar{b} - \bar{c}$.