Morley's Theorem and real algebraic geometry

Question

Consider the following attempt at a ``thought-free'' proof of Morley's Theorem.

Let $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3,y_3)$ denote three vertices of a generic triangle.

Let $(a_1,b_1)$, $(a_2,b_2)$ and $(a_3,b_3)$ denote three more points in the plane.

For all $\lbrace i,j,k\rbrace=\lbrace 1,2,3\rbrace$, we want to impose the condition that the line determined by $(x_i,y_i)$ and $(a_k,b_k)$ bisects the angle from $(a_j,b_j)$ to $(x_i,y_i)$ to $(x_j,y_j)$. Capturing this algebraically amounts to equating two squared cosines: $$ \frac{((x_i-a_k)(x_i-a_j)+(y_i-b_k)(y_i-b_j))^2} {((x_i-a_k)^2+(y_i-b_k)^2)((x_i-a_j)^2+(y_i-b_j)^2)}$$ $$ = \frac{((x_i-a_k)(x_i-x_j)+(y_i-b_k)(y_i-x_j))^2} {((x_i-a_k)^2+(y_i-b_k)^2)((x_i-x_j)^2+(y_i-x_j)^2)} $$ which simplifies first to $$ ((x_i-a_k)(x_i-a_j)+(y_i-b_k)(y_i-b_j))^2((x_i-x_j)^2+(y_i-x_j)^2)$$ $$ - ((x_i-a_k)(x_i-x_j)+(y_i-b_k)(y_i-x_j))^2((x_i-a_j)^2+(y_i-b_j)^2)=0. $$ Then expanding the left side and dividing out by the condition $$ -a_jy_i - b_jx_j + x_ib_j - x_ix_j + y_ix_j + a_jx_j$$ (of having $(a_j,b_j)$,$(x_i,y_i)$ and $(x_j,y_j)$ collinear) leaves $$C_{ijk}:=a_jy_i^3-x_i^3b_j+2x_ix_jy_ib_j-2x_ia_jy_ix_j-2a_ky_i^3+2x_i^3b_k+b_k^2x_ix_j+y_i^3x_j-x_i^3x_j$$ $$-a_k^2y_ix_j+2a_kx_ib_kx_j-2a_kx_jb_ky_i-2a_kb_kb_jy_i+2x_ja_kb_kb_j+2x_jb_ky_ib_j-2x_ja_ky_ib_j$$ $$-2x_jx_ib_kb_j+2a_ky_i^2b_j-a_k^2x_ib_j+2a_kx_i^2b_j+b_k^2x_ib_j-2b_k^2y_ix_i+2a_k^2x_iy_i+x_ja_k^2b_j$$ $$-x_jb_k^2b_j-a_jx_jb_k^2+a_jx_ja_k^2-a_ja_k^2y_i+a_jb_k^2y_i-2a_jb_kx_i^2-2a_jb_ky_i^2-2x_ja_kx_ib_j$$ $$+2a_jx_ja_ky_i-2a_jx_ja_kb_k-2a_jx_ja_kx_i+2a_jx_jb_ky_i+2a_jx_ib_kx_j+2a_ja_kx_ib_k+2a_kx_jy_i^2$$ $$-2y_i^2b_kx_j+b_k^2y_ix_j+2x_i^2a_kx_j+x_i^2y_ix_j-2x_i^2b_kx_j-x_ix_jy_i^2-a_k^2x_ix_j+x_i^2x_ja_j$$ $$-x_jy_i^2b_j+a_jy_ix_i^2-a_jy_i^2x_j+x_i^2b_jx_j-x_ib_jy_i^2-2a_kb_kx_i^2+2a_kb_ky_i^2-2a_ky_ix_i^2+2x_ib_ky_i^2\ .$$

Now for Morley's theorem we would like to conclude that points $(a_1,b_1)$, $(a_2,b_2)$ and $(a_3,b_3)$ form an equilateral triangle. We can easily write down polynomials in $a_1,b_1,a_2,b_2,a_3$ and $b_3$ all of whose vanishing would capture this. For example, it would suffice to prove the vanishing of $$I_{123}:=-2a_1a_2+a_2^2+2b_1^2-2b_1b_2+b_2^2+2a_1a_3-a_3^2-2b_1b_3+b_3^2\ .$$

Since we hope $I_{123}$ will vanish whenever $C_{123},C_{213},C_{312},C_{132},C_{231},C_{321}$ do, we look to find $I_{123}$ in (the radical of) the ideal $$C:=\langle C_{123},C_{213},C_{312},C_{132},C_{231},C_{321}\rangle.$$ Unfortunately, as one can see from the above, all the generators of $C$ vanish if $x_1=y_1=x_2=y_2=x_3=y_3=0$, and so the same must hold for every element of $C$. Nevertheless $I_{123}$ does not vanish on this condition.

The classical nullstellensatz leaves only one way out: the vanishing of $I_{123}$ doesn't follow from the vanishing of all the $C_{ijk}$ without the exploitation of some restriction on the range of the variables such as confining them to the reals.

Question: What insight does the machinery of real algebraic geometry shed on the shape of a proof of Morley's Theorem along the lines of this sketch?

Answer 1

This isn't a matter of real versus complex geometry -- the equations you give aren't enough to encode Morley's theorem over the reals. Let $z_1 z_2 z_3$ be our original triangle; I will use indices that are cyclic modulo $3$. Let $\ell^0_i$ be the trisector of the angle at $z_i$ which is closer to $z_{i-1}$ and let $m^0_i$ be the trisector which is closer to $z_i$. Let $\ell^1_i$ and $\ell^2_i$ be the rotations of $\ell^0_i$ by $120^{\circ}$ and $240^{\circ}$, and define $m^1_i$ and $m^2_i$ likewise.

For $r=1$, $2$ and $3$, let $(\delta_r, \epsilon_r)$ equal to one of $(0,0)$, $(1,2)$ or $(2,1)$. Let $c_r$ be the intersection of $m^{\epsilon_r}_r$ and $\ell^{\delta_{r+1}}_{r+1}$. Then $\angle z_r z_{r+1} c_r = \angle c_r z_{r+1} c_{r+1} = \angle c_{r+1} z_{r+1} z_{r+2}$ so your equations hold for $(x_r, y_r) = z_r$ and $(a_r, b_r) = c_r$.

Of the $81$ possible choices for $(\delta_r, \epsilon_r)$, I believe that the triangle $c_1 c_2 c_3$ is equilateral only for $54$ of them. I am basing this on Connes' proof of Morley's theorem, where a key hypothesis is that $$\sum_{r=1}^3 \angle c_{r-1} z_{r} c_{r} = \pm 60^{\circ} \ \mbox{not} \ 180^{\circ}.$$ Connes' rotation $g_r$ is a rotation around $z_r$ by angle $2 \angle c_{r-1} z_{r} c_{r}$, so what he says is that $g_1 g_2 g_3$ should be a nontrivial rotation.

It's easy enough to encode this condition algebraically. (If you have trouble with it I'll write more, but I bet you won't.) Once you do that, I think it should be easy to convert Connes proof into the sort of proof you are looking for. You'll also need to impose that certain things are nonzero, because Connes divides at times.