<s>[INST] Proving Altitudes of Triangle can never form a Triangle Prove that Altitudes of Triangle can never form a Triangle 
My try: we have altitudes proportional to reciprocals of sides of given triangle
Let $a,b,c$ are sides we have 
$$a+b \gt c $$
$$b+c \gt a$$ and $$c+a \gt b$$
Now if $\frac{1}{a}$, $\frac{1}{b}$ and $\frac{1}{c}$ are sides of another Triangle  we need to show that
$\frac{1}{a}+\frac{1}{b}$ Can never be Greater than $\frac{1}{c}$
Any hint?
 [/INST] The condition for a triangle is
$c(a+b) > ab$.
For the 3-4-5 triangle,
this is
$5(3+4) > 3\cdot 4$
which is true.
So the altitudes do form a triangle
in this case.
Note that you can write
the condition as
$\begin{array}\\
0
&\gt ab-c(a+b)\\
&= ab-c(a+b)+c^2-c^2\\
&= (a-c)(b-c)-c^2\\
\text{or}\\
c^2
&\gt (a-c)(b-c)\\
&= (c-a)(c-b)\\
\end{array}
$
This is true if
$c$ is the largest side
of the triangle,
so it is always true.
Therefore,
the altitudes can
always form a triangle.
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