Belatedly, sorry: now asked at MO.
Given a topological space $\mathcal{X}$ and finite-arity partial functions $f_i$ which $(1)$ are defined on a dense open subset of (the appropriate product of) $\mathcal{X}$ and $(2)$ are continuous on their domains, let the broad equational theory of $(\mathcal{X};f_1,…,f_n)$ be the set of all equations (in the sense of universal algebra) in (symbols standing for) the functions $f_1,…,f_n$ such that both sides are defined and equal for a dense open set of inputs.
For example, if we take $\mathcal{X}=\mathbb{R}^2$ with the usual topology, then the ternary orthocenter function $\mathsf{oc}$ is defined and continuous on a dense open subset of $\mathcal{X}^3$, and the equation $$\mathsf{oc}(\mathsf{oc}(x,y,z),y,z)=x$$ holds for all $(x,y,z)$ in a (slightly smaller) dense open subset of $\mathcal{X}^3$ and so is in the broad equational theory of the orthocenter, $\mathsf{B}_{\mathsf{oc}}$. On the other hand, there is some messiness: whether or not the equation $\mathsf{oc}(x,x,x)=\mathsf{oc}(x,x,x)$ is in $\mathsf{B}_{\mathsf{oc}}$ or not depends on exactly how we define the orthocenter, since if the orthocenter of a "triangle" with all vertices equal is undefined then this equation is not in $\mathsf{B}_{\mathsf{oc}}$.
For simplicity, I'll mostly shift attention to $\mathsf{B_{oc}^+}:=$ the deductive closure of $\mathsf{B}_{\mathsf{oc}}$ in standard equational logic.
I'm curious exactly how complicated $\mathsf{B}_{\mathsf{oc}}$ is (I'm also interested in any of the other standard triangle centers, especially the circumcenter, but I'll focus on the orthocenter for now). At a very coarse level this is known: $\mathsf{B_{oc}}$ is computable by Tarski-Seidenberg. On the other hand, on a finer-grained level things are much less clear. For example:
Is $\mathsf{B_{oc}^+}$ finitely based?
(That is, is there a finite set of equations whose deductive closure, in equational logic, is $\mathsf{B_{oc}^+}$?)
There is a natural candidate for a finite base for $\mathsf{B_{oc}^+}$ – namely, the symmetry equations $$\mathsf{oc}(x,y,z)=\mathsf{oc}(y,x,z)\quad\mbox{and}\quad\mathsf{oc}(x,y,z)=\mathsf{oc}(x,z,y)$$ in addition to the involution equation $\mathsf{oc}(\mathsf{oc}(x,y,z),y,z)=x$ introduced above – but I have no real evidence that there aren't any more complicated equations involving the orthocenter.
A couple side comments:
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The shift from $\mathsf{B}_{\mathsf{oc}}$ to $\mathsf{B_{oc}^+}$ reflects my own limited experience with partial structures; I'm also happy – indeed more happy – to get an answer re: the finite-basedness of $\mathsf{B}_{\mathsf{oc}}$ in the context of partial equational logic provided that there is actually a good notion of finite-basedness in such a context developed already.
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Re: motivation, this question is an outgrowth of this other question of mine. Ultimately I'd like to understand how "rare" various equations are in the appropriate "space" of continuous triangle center functions. Already figuring out how rare the involutive property above is has turned out to be surprisingly difficult, so I've started looking for other questions which might be simpler. The current question certainly has a lot more abstract baggage around it, but on the other hand is studying a single well-understood geometric function, so hopefully will be easier to address. (And yes, it's not about "rarity" at all, but that's because I'm easily distracted by shiny things.)
Best Answer
CW since this isn't my work:
Matt F. showed that the list of equations above does not axiomatize $T$, since $X(4)$ satisfies an additional equation which $X(74)$ does not. Meanwhile, at MO I've separately asked about finite basedness.