Draft:Aguilera-Brocard Triangles



The Aguilera-Brocard triangles are triangles of equal area that are formed with the Brocard points and three triangle centers on the Brocard axis. it derives its name from the work of Manuel Aguilera, a mathematics professor from Honduras, published in.

History
In 1881, French mathematician Henri Brocard publishes an article in the French Association for the Advancement of Science, giving rise to Brocard geometry by introducing the concepts of the Brocard circle and Brocard points. At the beginning of the 20th century American Geometer Roger Arthur Johnson showed that the Brocard points are symmetric with respect to the diameter of the Brocard circle. In the more recent past, all points created from a triangle are known as triangle centers following the publication of the Encyclopedia of Triangle Centers in the late 1990s by American Mathematician Clark Kimberling. As history progressed, several points, apart from the circumcenter and Lemoine point, are found along the Brocard axis, which later became known as Kimberling centers in the Brocard axis. The latter is crucial for the formation of the Aguilera-Brocard Triangles, as every Kimberling center on the Brocard axis aligns symmetrically with the Brocard points. These triangles can be expressed as:

$$R^2(2Cos[2w] - 1)Tan[w] \over (2*S)$$

The area of $$\bigtriangleup{ABC} = 1$$, Referring to the Brocard points as Ω1 and Ω2 we can designate two points $$P$$ and $$Q$$, on the Brocard axis with the aforementioned area. Let:

$$P = O + pL $$ and $$Q = O + qL$$, where $$O = X(3)$$ and $$L = X(6)$$. The area of Ω1, $$P$$ and $$Q$$ will be the area of Theorem 1.2 if

$$p =$$$$(1 + 3q) \over (1 - q)$$ and $$q = $$ $$(p - 1)\over (3 + p)$$. From the aforementioned equations for $$P$$ and $$Q$$, it can derive the following pairs of points {$${p, q}$$}, where each number in the square brackets represents a triangle center

Properties
In 2023, the Australian Mathematician Elias M Hagos found an interesting result about the Aguilera-Brocard Triangles, as mentioned below:

Theorem 1. The Triangle Centers of the Aguilera-Brocard Triangles in the Brocard Axis are found in the Stothers Quintic (Q12)

Finally, Greek Mathematician Antreas Hatzipolakis discovered the following property:

Theorem 2. The isogonal conjugates of the triangle centers within the Aguilera-Brocard Triangles, situated along the Brocard axis, are found on the Kiepert circumhyperbola.