Problem
GEO-B3-M05-P024 Steiner Point and the Brocard Diameter
Let \(A_1B_1C_1\) be the Brocard triangle of \(ABC\), and let \(S\) be the intersection point of the lines through \(A,B,C\) respectively parallel to \(B_1C_1,C_1A_1,A_1B_1\). Prove that \(S\) lies on the circumcircle of \(ABC\), and the Simson line of \(S\) is parallel to the Brocard diameter \(OK\).
C. Hint 1. Use that the Brocard triangle is tied to the similarity circle.
D. Hint 2. For the direction of the Simson line, use corresponding directions.
The vertices of the Brocard triangle are the constant points of three similar figures built on \(BC,CA,AB\). Therefore the lines through \(A,B,C\) parallel to the opposite sides of \(A_1B_1C_1\) are corresponding lines of these figures. By the similarity-circle property, their intersection \(S\) lies on the corresponding similarity circle; in this configuration, it is the circumcircle of \(ABC\).
Now \(S\) lies on the circumcircle, so it has a Simson line. The direction of this line is determined by the same corresponding lines, and the fixed direction in the similarity circle is the diameter \(OK\). Hence the Simson line of \(S\) is parallel to the Brocard diameter \(OK\).
This links Module 4 with the present module.