I've a question regarding the following situation :
Given (for instance) a triangle $ABC$ and two points $M$ and $N$ on $(AB)$ and $(CD)$ respectively such that $(MN) \not \parallel (BC)$.
($k$ and $h$ are given such that $\vec{AM} = k \vec{AB}$ and $\vec{AN} = h\vec{AC}$)
If $J$ denotes the point of intersection of lines $(BC)$ and $(MN)$, it is asked to determine the real number $\alpha$ in terms of $k$ and $h$ such that
$$\vec{BJ} = \alpha \vec{BC}$$
My approach would be :
Determine the equations of the two lines $(MN)$ and $(BC)$ in the referential $(A, \vec{AB}, \vec{AC})$ and solve the system of equations to get the coordinates of $J$ first then do what is required to get the $\alpha$.
I wonder whether there is a better / shorter approach to tackle this problem.
Thanks.
The answer depends on what is the information given, what lengths to write $\alpha$ from.
Menelaus' theorem allows you to express $\alpha$ as a function of the ratios that $M$ and $N$ divide the other sides of the triangle, and the side $BC$.