Equational theory of the orthocenter Previously asked at MSE:
Briefly speaking, I'm looking for a description of the equational theory of the orthocenter function, $\mathsf{orth}$. By $\mathsf{orth}$ I mean the (partial) function sending a triple $(a,b,c)$ of noncollinear points in $\mathbb{R}^2$ to the orthocenter of the resulting triangle.
Since the orthocenter function is partial - if $x,y,z$ are collinear then $\mathsf{orth}(x,y,z)$ doesn't exist - a bit of care is needed to make this precise. I'm not familiar with partial universal algebra, so I'll adopt the following brute-force fix: let $T$ be the deductive closure, in the sense of standard equational logic, of the set of all equations in $\mathsf{orth}$ which are true on a dense open set of inputs. Basically, this lets us ignore partiality issues. For example, the equation $$\mathsf{orth}(x,x,x)=\mathsf{orth}(x,x,x)$$ is bonkers for any input, but is a tautology in the sense of standard equational logic, so is in $T$.
There is a natural candidate for an equational axiomatization of $T$:

*

*The symmetry equations $$\mathsf{orth}(x,y,z)=\mathsf{orth}(y,z,x)\mbox{ and }\mathsf{orth}(x,y,z)=\mathsf{orth}(x,z,y),$$ and


*the involutivity equation $$\mathsf{orth}(x,y,\mathsf{orth}(x,y,z))=z.$$

Question: Do these three equations in fact yield an axiomatization of $T$? If not, is $T$ finitely based at all? (EDIT: I've gone ahead and asked the finite basedness question separately.)

(It may be helpful, towards a negative solution, to consider the analogues of $T$ corresponding to the two other known "involutive" triangle center functions, $X(74)$ and $X(1138)$. Since each of these correspond to functions satisfying the above equations in the appropriate sense, an equation "true" about the orthocenter but "false" about $X(74)$ or $X(1138)$ would answer the question. However, I don't see such an equation.)
 A: $\newcommand{\o}[0]{\mathsf{orth}}$No, these equations do not yield the complete theory of the orthocenter.
The identity
$$\o(\o(t,u,v),\o(t,u,w),u) = \o(\o(t,u,v),\o(t,v,w),v)$$
holds for the orthocenter (X(4)) but not for X(74) (the isogonal conjugate of the Euler infinity point), even though both satisfy involutory identities like
$$v=\o(t,u,\o(t,u,v))$$
I found the first identity and tested both using Mathematica, which can set up the functions quickly as follows:
avg[a_, b_, c_, u_, v_, w_] := (a u + b v + c w)/(a + b + c);    
bary[f_, u_, v_, w_] := avg[f[u,v,w], f[v,w,u], f[w,u,v], u, v, w];
cosA[u_, v_, w_] := (v-u).(w-u) / Sqrt[((v-u).(v-u)) ((w-u).(w-u))];
f[u_, v_, w_] := 1/((u-v).(u-w));
g[u_, v_, w_] := Sqrt[(v-w).(v-w)] / (cosA[u,v,w] - 2 cosA[v,w,u] cosA[w,u,v]);
center4[u_, v_, w_]  := bary[f, u, v, w];
center74[u_, v_, w_] := bary[g, u, v, w];

Then the following code tests the identity for the orthocenter algebraically, and for the X(74) center numerically:
Algebra = {t -> {tx, ty}, u -> {ux, uy}, v -> {vx, vy}, w -> {wx, wy}};
Example = {t -> {0, 1}, u -> {2, 4}, v -> {-3, 5}, w -> {-2, 1}};  
FourVariable[c_] := c[c[t,u,v], c[t,u,w], u] == c[c[t,u,v], c[t,v,w], v];
{FourVariable[center4] /. Algebra // Simplify, FourVariable[center74] /. Example}

and the tests return True and False respectively.
