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The (squares of the) lengths of the median, angle bisector, and altitude of an $a$-$b$-$c$ triangle with base $a$ are given by the following:

$$\begin{align} d_{\text{med}}^2 &= \frac14\left( - a^2 + 2 b^2 + 2 c^2 \right) \tag{1}\\[4pt] d_{\text{bis}}^2 &= \frac{b c}{(b+c)^2}(a+b+c)(-a+b+c) \tag{2}\\[4pt] d_{\text{alt}}^2 &= \frac{1}{4a^2}(a+b+c)(-a+b+c)(a-b+c)(a+b-c) \tag{3}\\[4pt] \end{align}$$

(These can be derived using, for instance, Stewart's formula.)

I'm wondering if there's a unified formula for $(1)$, $(2)$, $(3)$, with the instances distinguished by some auxiliary parameter. In other words,

  • Is there a function $f(a,b,c,n)$ such that $$f(a,b,c,n_\text{med})=d_\text{med}\qquad f(a,b,c,n_\text{bis})=d_\text{bis}\qquad f(a,b,c,n_\text{alt})=d_\text{alt}$$ for some $n_\text{med}$, $n_\text{bis}$, $n_\text{alt}$ that are independent of $a$, $b$, $c$?

  • And, is it possible that these particular values of $n$ can be taken as (consecutive) integers?

  • And, further, might other special cevians be associated with other specific (possibly integer) values of $n$?

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  • $\begingroup$ How are you associating $n=0, 1, 2$ to the altitude, angle bisector, and median? And what are the "interesting results" associated with $n=-\infty, -1, \infty$, and what justifies the association of those value of $n$ to those results? (Hmmm ... After an edit, I read that the associations with $n=-1,0,1,2$ aren't necessarily those you give.)... In any case, as you note, the formula for the length of a cevian is already known; it either is or isn't an analogue of the "generalized mean" in the way you want. You seem to be asking us to read your mind about how this is supposed to work. $\endgroup$ – Blue Jun 3 at 3:33
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    $\begingroup$ BTW: Instead of posting a drafty question and then making a zillion edits, you should use the sandbox in Meta to get your question to a reasonably-stable form. Rampant "live" editing makes it impossible to know when a question is ready for proper consideration. $\endgroup$ – Blue Jun 3 at 3:38
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    $\begingroup$ The description is quite confusing in this regard. Here's my understanding of what you're asking ... "Is there a function $f(a,b,c,n)$ that, for a specific $n=n_{\text{alt}}$, yields the length of the altitude of the $a$-$b$-$c$ triangle (with base $a$)? and likewise, for $n=n_{\text{bis}}$ and $n=n_{\text{med}}$, yields the lengths of the angle bisector and median? And is it possible that these particular values of $n$ can be taken as (consecutive) integers? And, further, might other special cevians be associated with other specific (possibly integer) values of $n$?" $\endgroup$ – Blue Jun 3 at 3:57
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    $\begingroup$ I suggest editing. While our comment-discussion may help clarify your intent, people shouldn't have to consult the comments (and very often don't) to understand a question. ... Speaking of which: I'm getting the "avoid extended discussions" warning. I will likely not be responding further. Good luck! $\endgroup$ – Blue Jun 3 at 4:21
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    $\begingroup$ I went to fix some formatting in the question, but couldn't seem to stop myself from significantly revising the question. :) Since you used my commented suggestion for your own revision, I didn't think you'd mind. In any case ... Please double-check that I've properly captured your intention. $\endgroup$ – Blue Jun 3 at 11:36

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