Brocard Triangles
Given triangle Δ A_1 A_2 A_3, let the point of intersection of A_2 Ω and A_3 Ω' be B_1, where Ω and Ω' are the Brocard points, and similarly define B_2 and B_3. Then Δ B_1 B_2 B_3 is called the first Brocard triangle, and is inversely similar to Δ A_1 A_2 A_3. It is inscribed in the Brocard circle. Let c_1, c_2, and c_3 be the circles through the vertices A_2 and A_3, A_1 and A_3, and A_1 and A_2, respectively, which intersect in the first Brocard point Ω. Similarly, define c_1^, , c_2^, , and c_3^, with respect to the second Brocard point Ω'. Let the two circles c_1 and c_1^, tangent at A_1 to A_1 A_2 and A_1 A_3, and passing respectively through A_3 and A_2, meet again at C_1, and similarly for C_2 and C_3. Then the triangle Δ C_1 C_2 C_3 is called the second Brocard triangle.
