Find ratio of segments in triangle

Asked Jan 14 '22 at 11:34

Active Jan 14 '22 at 11:34

Viewed 58 times

We are given a triangle $ABC$ and points $D$ and $E$ on $AB$, such that $AE = ED = DB$.

$F$ and $G$ are arbitrary points on $AC$ and $BC$ respectively. We want to find the product of ratios $\frac {EH}{HF} \cdot \frac{FM}{MB}$.

I have tried drawing several parallels and applying Thales theorem and then multiply but I am not getting anywhere.

I also tried to use the fact that $FE$ is the median in triangle $AFD$ and $FD$ is the median in triangle $EFb$ but still nothing.

I also attempted to extend $CB$ to take a segment $BI = BC$, so that in triangle $AIC$, $AB$ will be the median and if we extend $IB$, it will bisect the side $AC$. But I still don't see anything.

Geogebra gives the result $\frac {1}{3}$.

Any ideas please? (this is not homework or anything. I found it in a Euclidean Geometry group).

asked Jan 14 '22 at 11:34

Pradeep Suny

1,603

Try to make smaller drawings!! – Carlos Lopez Jan 14 '22 at 11:37
1

Apply Menelaus's theorem in $\triangle EBF$ with secant $MA$ intersecting sides $BF$ and $EF$ internally and side $BE$ externally.
So, $ \displaystyle \frac{BA}{AE} \cdot \frac{EH}{HF} \cdot \frac{FM}{MB} = 1$
– Math Lover Jan 14 '22 at 11:47
1

https://en.wikipedia.org/wiki/Menelaus%27s_theorem – Math Lover Jan 14 '22 at 11:49
1

@MathLover got it, thank you very much! – Pradeep Suny Jan 14 '22 at 12:04
@PradeepSuny, you can answer your own question and accept your answer to remove this from unanswered list. – ACB Jan 15 '22 at 10:37

Find ratio of segments in triangle

0 Answers0