Brocard Triangles

Let the point of intersection of $A_2\Omega$ and $A_3\Omega'$ be $B_1$, where $\Omega$ and $\Omega'$ are the Brocard Points, and similarly define $B_2$ and $B_3$. $B_1B_2B_3$ is the first Brocard triangle, and is inversely similar to $A_1A_2A_3$. It is inscribed in the Brocard Circle drawn with $OK$ as the Diameter. The triangles $B_1A_2A_3$, $B_2A_3A_1$, and $B_3A_1A_2$ are Isosceles Triangles with base angles $\omega$, where $\omega$ is the Brocard Angle. The sum of the areas of the Isosceles Triangles is $\Delta$, the Area of Triangle $A_1A_2A_3$. The first Brocard triangle is in perspective with the given Triangle, with $A_1B_1$, $A_2B_2$, and $A_3B_3$ Concurrent. The Median Point of the first Brocard triangle is the Median Point $M$ of the original triangle. The Brocard triangles are in perspective at $M$.

Let $c_1$, $c_2$, and $c_3$ and $c_1'$, $c_2'$, and $c_3'$ be the Circles intersecting in the Brocard Points $\Omega$ and $\Omega'$, respectively. Let the two circles $c_1$ and $c_1'$ tangent at $A_1$ to $A_1A_2$ and $A_1A_3$, and passing respectively through $A_3$ and $A_2$, meet again at $C_1$. The triangle $C_1C_2C_3$ is the second Brocard triangle. Each Vertex of the second Brocard triangle lies on the second Brocard Circle.

The two Brocard triangles are in perspective at $M$.

See also Steiner Points, Tarry Point

References

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 277-281, 1929.