Consider a barycentric coordinate system with points $v_0,v_1,v_2$ and $x=\alpha_1 v_1 + \alpha_2 v_2$ and $y=\beta_0 v_0 + \beta_2 v_2$.
Find the intersection $s$ of the lines through $x$ and $y$ and through $v_0$ and $v_1$.
I have the parametric forms:
$g_1(t)=x+t(y-x)=(0,\alpha_1,\alpha_2)+t((\beta_0,0,\beta_2)-(0,\alpha_1,\alpha_2))=(0,\alpha_1,\alpha_2)+t(\beta_0,-\alpha_1,\beta_2-\alpha_2)$ $g_2(u)=v_0+u(v_1-v_0)=(1,0,0)+u((0,1,0)-(1,0,0))=(1,0,0)+u(-1,1,0)$
For the second form I thought that $v_0$ is exactly one times itself and zero times the other points and the same for $v_1$.
When I set the equations equal and solve by component I don't really get a solution so I'm not sure if my approach is correct. Thanks for any help!