# Geometric problem about ratios of line segments: How to transform the limiting case method to a rigorous answer?

In $$\triangle ABC$$, $$D, E, F, G$$ are points on the sides of the triangle such that $$BD:DE:EC=1:2:3$$, $$AF:CF=1:1$$, and $$AG:BG=2:3$$. Find the ratio $$FH:DH$$.

My classmate has come up with a brilliant way to do this. He argued that if we let $$AC=1, AB=5,BC=6,$$

The triangle will collapse into a straight line.

We can observe directly

$$FH:DH=2.5:2=5:4$$

Despite the lack of rigor, this method successfully computes the right solution. My question is:

Can we transform the limiting case method (as I would call it) to a rigorous answer?

Yes, for this case, basically it's equivalent to projecting everything onto $$BC$$ along direction of the line $$GHE$$. Since we're looking for a ratio, it doesn't change when we perform projection.