In $\Delta ABC$, $BA=20,BC=25,$ and segment $BM$ is a median. $R$ is any point on segment $BC$. $S$ lies on segment $BA$ with $BS=\frac35 BR$. $SR$ and $BM$ interesect at $T$. Find the ratio $ST:RT$.
I believe I can generalize the case if I can first figure out the specific case where $R$ is the same as the point $C$. I've included the diagram here, where I've constructed parallel lines through $A$ and $C$ that are parallel to $BM$. I've called the intersection of $SC$ and the parallel line through $A$ point $D$.
For this case, I know that $BS=15$ and $SA=5$. I think my goal will probably be that I need to show that $\Delta ASD$ is similar to one of the other triangles so that I can compare the side lengths, but I'm not exactly sure which ones/how.
Divide the figure into regions with colors as shown.
Note (1) the grey coded parts are equal in area; (2) [green + blue] = .... = [yellow + pink]
$\dfrac {[green]}{[green + blue]} = \dfrac {3k}{20}$
Setup a similar ratio for the yellow and pink.
Perform a suitable operation to those two ratios such that the [green + blue], [yellow + pink], and 'k' terms can be eliminated altogether.
The required is equal to $\dfrac {[green]}{[pink]}$.