Ratio of the heights of triangle is given, determine the sides of triangle. Can someone help me with this excersize?
The ratio of heights to the sides of the triangle is v_{a}:v_{b}:v_{c}=12:5:8. What are the length of the sides of triangle? (a,b,c=?) 
Thank you!
I tried using formula for area of triangle, e.g. P=(av_a)/2 and P=(bv_b)/2 and then I would get b=12/5 a and I did that for other combinations but did not come up with anything.

Yes, your idea works!
Since $$S=\frac{ah_a}{2}=\frac{bh_b}{2}=\frac{ch_c}{2},$$ we obtain:
$$\frac{1}{a}:\frac{1}{b}:\frac{1}{c}=12:5:8$$ or
$$a:b:c=\frac{1}{12}:\frac{1}{5}:\frac{1}{8}.$$
We need to check that $$\frac{1}{12}+\frac{1}{8}>\frac{1}{5},$$ of course. 
For minimal natural's $a$, $b$ and $c$ we obtain $(10,24,15).$ 
