Draft:Aguilera-Brocard triangles

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In plane geometry, Aguilera-Brocard triangles are a set of triangles that arise from the properties of the Brocard circle and Brocard points in a triangle. The Brocard circle, defined by Henri Brocard in 1881, is the circle with diameter $OL$, where $O$ is the circumcenter and $L$ is the Lemoine point (the intersection of the symmedians) of the triangle. The Brocard points, ${\mathrm{\Omega }}_1$​ and ${\mathrm{\Omega }}_2$​, are symmetric with respect to the diameter $OL$ of the Brocard circle. This triangles was discovered by Professor Manuel M., Aguilera in 2023

Ilustration of the Aguilera-Brocard triangles in the Brocard circle

Johnson's Theorem

During the year 1929, the mathematician Roger Arthur Johnson published several findings in his book Johnson's Modern Geometry, among those findings was a theorem that mentioned that four triangles have identical areas on the Brocard circle as described now.

Theorem — Let ${\mathrm{\Omega }}_1$​ and ${\mathrm{\Omega }}_2$​ be the Brocard points, $O$ the circumcenter, $L$ the Lemoine point, and ​$X_{182}$ the center of the Brocard circle. it will be fulfilled that triangles $△$${\mathrm{\Omega }}_1$​​​$X_{182}$$O$, $△$${\mathrm{\Omega }}_2$​​​$X_{182}$$O$, $△$${\mathrm{\Omega }}_1$​​​$X_{182}$$L$, and ${\mathrm{\Omega }}_2$​​​$△$$X_{182}$$L$ will all have the same area.

This theorem forms the basis for the Aguilera-Brocard triangles.

Aguilera-Brocard triangles

Theorem — The Aguilera-Brocard triangles are pairs of triangles with equal area formed by the Brocard points (${\mathrm{\Omega }}_1$​ and ${\mathrm{\Omega }}_2$​) and two triangle centers ($P$ and $Q$) located on the Brocard axis. The Brocard axis is the line connecting the circumcenter $O$ and the Lemoine point $L$.

Proof. Let the pair Aguilera-Brocard triangles be defined as $T_1=$ $△$${\mathrm{\Omega }}_1$​​​$P$$Q$ and $T_2=$ $△$${\mathrm{\Omega }}_2$​​​$P$$Q$. The Brocard points (${\mathrm{\Omega }}_1$​ and ${\mathrm{\Omega }}_2$​) are symmetric with respect to the Brocard axis. This symmetry implies that the perpendicular distances from ${\mathrm{\Omega }}_1$​ and ${\mathrm{\Omega }}_2$ ​ to any line lying on the Brocard axis are equal and the line $PQ$ lies entirely on the Brocard axis, and thus serves as the common base for both triangles $T_1$ and $T_2$. The height of $T_1$ and $T_2$ is the perpendicular distance from ${\mathrm{\Omega }}_1$​ and ${\mathrm{\Omega }}_2$ respectively to the line $PQ$. By the symmetry of ${\mathrm{\Omega }}_1$​ and ${\mathrm{\Omega }}_2$ with respect to the Brocard axis, these perpendicular distances are equal. Now, the area of these triangles is

$𝒜(T_1)=\frac{1}{2}\times PQ\times h_1$ and $𝒜(T_2)=\frac{1}{2}\times PQ\times h_2$ where h_1 and h_2 are the heights of the pair of Aguilera-Brocard triangles congruent to each other. Since $h_1=h_2$ ​(by symmetry), it follows that $𝒜(T_1)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}𝒜(T_2)$.

Key Properties

• The Aguilera-Brocard Triangles are pairs of triangles with equal area, formed by the Brocard points and two points on the Brocard axis.
• The Brocard axis is a central line in the triangle, and the points $P$ and $Q$ are chosen such that the area of $T_1=$ $△$${\mathrm{\Omega }}_1$​​​$P$$Q$ and $T_2=$ $△$${\mathrm{\Omega }}_2$​​​$P$$Q$ is preserved.

Applications and Extensions

• The Aguilera-Brocard triangles can be extended to points on the Stothers quintic (Q012), which is the locus of points $P$ such that the line $PQ$ is orthogonal to the line $W_1{W}_2$, where $W_1$ and $W_2$ are the Brocardians of $P$, and $Q$ is the center of the circle $P{W}_1{W}_2$.
• The quintic passes through several notable points, including $X(2)$, $X(6)$, $X(7)$, $X(13)$, $X(673)$, and $X(694)$.