Question 1074278: In triangle RST, U lies on TS with TU:US= 2:3. M is the midpoint of RU. TM intersects RS in V. Find the ratio of RV:RS.
I was able to solve it with trig, and I got 2:7. It was meant to be solved with basic geometry, but I haven't be able to think of a way.
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! Maybe by now you have been able to think of a way.
I could not think of a way late last night,
but I woke up with a clearer mind this morning,
so I added line MW, parallel to ST:
 MW is part of two new triangles similar to two triangles that were there before.
Now that I look at the drawing,
I realize it has TU:RS = 3:2 .
A drawing is not necessary to support the reasoning,
but even an inaccurate drawing helps
as a reminder of the meaning of each point.
RWM is similar to RSU.
MVW is similar to TVS.
The "scale down factors" between the similar triangles
lead to a way to calculate the lengths of RW and VW
as fractions of the length pf RS,
and that allows to calculate, as a fraction of the length of RS,
the lengths of  .
Since  and  ,
So,
 and  .
Since M is the midpoint of RU,
and WM is parallel to ST,
RWM is not only similar to RSU;
it is a  scale version of RSU.
So, not only is MW the midsegment of RSU,
with  ,
but also  ,
and of course,  too.
 .
Since MVW is similar to TVS,
and  ,  .
Finally,
 -->  .
Form there,
 -->  -->  -->  ,
and  -->  --> 
NOTE:
I am not sure if the reasoning above is the shortest way to the answer.
Maybe I should have drawn a line through M parallel to RS instead.
With nothing else to do, I would explore other angles to this problem
(and draw a better drawing).
However, if you are still looking for a geometry-based answer,
a less than perfect answer is better than no answer.
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