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 $n$?