I'm writing some notes about geometry in $\Bbb R^2$ and I have given these definitions:
- Given two different points $A$ and $B$, $X$ is a point of the segment $AB$ iff there exists some $t\in[0,1]$ such that $\overrightarrow{AX}=t\overrightarrow{AB}$. If $t$ isn't $0$ or $1$, we say that $X$ is an interior point of the segment.
- Given three non-colinear points $A$, $B$, $C$, the triangle $ABC$ is the union of the segments $AB$, $BC$ and $AC$ (the sides of the triangle) and the set of points $P$ such that there exists some interior point $X$ of the segment $AB$ such that $P$ is an interior point of the segment $CX$.
Now I want to prove the consistency of the definition when we change the side, that is, I want to prove that if $P$ meets the given condition, then there exists some interior point $Y$ in the segment $BC$ such that $P$ is an interior point of $AY$.
I have been "playing" with vectors for hours, avoiding using the coordinates of the vectors explicitly. I find two reasons. First: this should work at every Euclidean space $\Bbb R^n$, and second, using the coordinates, equations of sides, etc, would make the proof somewhat horrible. But I couldn't catch the point so far... quite literally.
Note: moduli, dot product and angles are defined, but I couldn't figure out how to use them.
Any ideas?