The third Brocard triangle is Gibert's term for the isogonal conjugate of the first Brocard triangle.
It has trilinear vertex matrix
![[b^2c^2 ab^3 ac^3; a^3b a^2c^2 bc^3; a^3c b^3c a^2b^2].](/images/equations/ThirdBrocardTriangle/NumberedEquation1.svg) |
Three of the 36 cataloged six-piece triangle dissections of a generic non-right triangle into similar triangles have interior junctions at the three vertices of the third Brocard triangle. Each layout has four boundary cut points on a circle. The three circles are distinct, and their centers lie on the Brocard axis. The six endpoints of the three straight-through cuts instead lie on a common conic section by Carnot's conic theorem. This conic section is generically not a circle (E. Pegg, Jr., pers. comm., Oct. 3, 2026).
Pegg, E. Jr. Mathematical Games. Episode 46: "Sporadic Similar
Triangles." Oct. 22, 2026.