Construct any triangle ABC and, given any point P (for now inside the triangle), extend rays from the vertices of the triangle through P, resulting in the following diagram:
Proposition: 
Proof: This proof depends on the construction of lines, one through A and one through C, both parallel to ray BF:
We next notice 5 pairs of congruent triangles:
And finally,
Now, since corresponding parts of congruent triangles are congruent, we gleen the following proportions:
Then 
.
And thus 
.
This proposition is actually true for some values of P outside of triangle ABC. To explore this you may click here to open a GSP worksheet. Move P around and notice how this ratio stays constant, as well as when it becomes undefined.