I am trying to prove the following statement:
P is a point of the segment AB $\iff \vec{CP} = \alpha \vec{CA} + \beta \vec{CB}$
where $A,B,C$ are arbitrary points, with $A \neq B$ , and $\alpha, \beta > 0$, $\alpha + \beta = 1$. Using the following result from the previous exercise, I have managed to prove the direct implication:
Given four points $A,B,C$ and $P$ such that $\vec{AP} = \lambda \vec{PB}$, represent the vector $\vec{CP}$ in function of $\vec{CA}, \vec{CB}$ and the parameter $\lambda$. So $\vec{CP} = \frac{\vec{CA} + \lambda\vec{CB}}{\lambda +1}$
And so $\vec{CP} = \alpha\vec{CA} + \beta\vec{CB}$, where $\alpha = 1/(\lambda+1)$ and $\beta = \lambda/(\lambda+1)$
But I can't prove the other side of the implication. Can someone give me a tip?
Hint: Write $\overrightarrow{CP}=\overrightarrow{AP}-\overrightarrow{AC},\overrightarrow{CA}=-\overrightarrow{AC},\overrightarrow{CB}=\overrightarrow{AB}-\overrightarrow{AC}$, and simplify.