In connection with a question asked in chat, I thought about this:
Question: What are sufficient and necessary conditions for three numbers $h_a$, $h_b$, $h_c$ to be heights of a triangle?
I learned from the Wikipedia article "List of triangle inequalities § Altitudes" (current revision) that:
The reciprocals of the altitudes of any triangle can themselves form a triangle: $$\frac{1}{h_a}<\frac{1}{h_b}+\frac{1}{h_c}, \quad \frac{1}{h_b}<\frac{1}{h_c}+\frac{1}{h_a}, \quad \frac{1}{h_c}<\frac{1}{h_a}+\frac{1}{h_b}.$$
Wikpedia gives as a reference: Mitchell, Douglas W., "A Heron-type formula for the reciprocal area of a triangle", ''Mathematical Gazette'' 89 (November 2005), 494.
Question: The above three inequalities giva a necessary condition - are they also a sufficient condition? (I.e., if we're given $h_a$, $h_b$, $h_c$ that fulfill these three inequalities, is there a triangle with altitudes of those lengths?)
I guess the answer is yes. For example if I look at the result from the blog post called Altitude Reciprocals (Wayback Machine) in The Lost Math Lessons blog (by David Vreken), I read that:
The reciprocals of the altitudes of any triangle can themselves form a triangle that is similar to the original triangle.
So it seems that it would be enough to take the triangle with the sides $\frac1{h_a}$, $\frac1{h_b}$, $\frac1{h_c}$ and then to scale it correctly. But maybe there are some other, more straightforward ways to see this.
This seems a very natural question, so I expected to find something about this on Mathematics Stack Exchange. But I did not succeed. (It is certainly possible that the same question has been asked before, but I failed to find a duplicate.)