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Given $\triangle QUS$, let R be a point on $\overline{QS}$ such that $QR = RS$ and T be a point on $\overline{US}$ such that $ST=3UT$. Also, note that $\overline{UR}$ and $\overline{QT}$ intersect at V.

Prove that $\frac{[UVT]}{[URS]}$=$\frac{1}{10}$.

Note: When I presented this to a friend, he suggested that I use mass points. However, I am not yet familiar with the topic so I am hoping to solve this using more elementary concepts such as similar triangles or area ratios.

Edit: Thanks for your comment, @Ivan Kaznacheyeu. There were indeed typos to my question. I've edited it for clarity.

chuckong083608
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    So what did you try? – Math Lover Jul 21 '22 at 04:13
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    I consider mass points as elementary. It’s a basic physics concept, I highly recommend learning it. A quick Google/YouTube search will give you results. – TheBestMagician Jul 21 '22 at 05:20
  • I tried some equations based same base - same height principles like [UQT] = 3 [TQS] and [QUR] = [SUR] but I don't know how to connect both of them.

    I also can't seem to determine any similar triangles.

    I know that the ratio of QV/VT seem to be an essential part of the solution since it acts like a hinge but I don't know where to start.

     – chuckong083608 Jul 21 '22 at 05:34
  • @TheBestMagician all right. seems like the more viable option. Will do so. Thanks. – chuckong083608 Jul 21 '22 at 05:35
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    There are two possible positions of $T$ in the problem, but no one gives result $\frac{1}{10}$. Maybe there is problem in problem statement. To find solution, which OP seeks (using similar triangles and ratio of areas of triangles having common altitude), one can use point W on line US, such that RW is parallel to QT. – Ivan Kaznacheyeu Jul 21 '22 at 09:17
  • @IvanKaznacheyeu I looked at it again and you are indeed correct about the typos. Sorry about that. I've edited the question for clarity. Will also try your suggestion. Cheers! – chuckong083608 Jul 21 '22 at 12:51
  • +1, thank you for your prompt actions and good question. – Sarvesh Ravichandran Iyer Jul 21 '22 at 12:56
  • Now question is correct. My comment about $W$ works for corrected question. Try to use it. If there will be still problem with solution, I can add the answer. – Ivan Kaznacheyeu Jul 21 '22 at 13:00
  • @IvanKaznacheyeu Thanks! I think I've gotten it. Posting it for reference. – chuckong083608 Jul 21 '22 at 14:25

1 Answers1

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Thanks to @Ivan Kaznacheyeu for the hint.

Assign a point W on $\overline{US}$ such that $\overline{RW} || \overline{QT}$.

Note that $\triangle SRW \sim \triangle SQT$. Therefore $TW=WS$.

Also, note that $\triangle UVT \sim \triangle URW$. Therefore $\frac{UT}{UW} = \frac{2}{5}=\frac{UV}{UR}$.

Therefore, $\frac{[UVT]}{[URS]}=\frac{(UV)(UT)}{(UR)(US)}=(\frac{UV}{UT})(\frac{UT}{US})=(\frac{2}{5})(\frac{1}{4})=\frac{1}{10}$

chuckong083608
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