What is the ratio of the sides of a triangle in which the sum of the lengths of two altitudes, taken in pairs, is in the ratio 5:7:8?
Let the heights in the first two sums be denoted by $m_{a}$ and $m_{b}$, and $m_{b}$ and $m_{c}$, respectively. Then the third is equal to $m_{a}+m_{c}$, and according to the problem,

\[
\begin{aligned}
m_{a}+m_{b} & =5 x \\
m_{b}+m_{c} & =7 x \\
m_{a}+\quad m_{c} & =8 x
\end{aligned}
\]

where $x$ is an unknown positive number. Add the three equations and divide the resulting sum by 2:

\[
m_{a}+m_{b}+m_{c}=10 x
\]

If we subtract the above equations from this, we get that

\[
m_{a}=3 x, \quad m_{b}=2 x, \quad m_{c}=5 x
\]

If we multiply these quantities by the lengths of the sides opposite the respective heights, we get the same value:

\[
a m_{a}=b m_{b}=c m_{c}=2 t
\]

that is,

\[
3 a x=2 b x=5 c x=2 t
\]

where $t$ denotes the area of the triangle. According to the last relationship,

\[
a=\frac{2 t}{3 x}, \quad b=\frac{2 t}{2 x}, \quad c=\frac{2 t}{5 x}
\]

thus the ratio of the sides of the triangle is:

\[
a: b: c=\frac{2 t}{3 x}: \frac{2 t}{2 x}: \frac{2 t}{5 x}=10: 15: 6
\]