Problem
GEO-B3-M05-P019 Two Brocard Points
Let \(P\) be the first Brocard point of triangle \(ABC\). Prove that its isogonal conjugate is the second Brocard point.
C. Hint 1. Write the angle equalities for the first Brocard point.
D. Hint 2. Reflect \(AP,BP,CP\) across the angle bisectors.
Let \(\angle ABP=\angle BCP=\angle CAP=\varphi\). Reflect \(AP,BP,CP\) in the angle bisectors of \(A,B,C\). By trigonometric Ceva, the three reflected lines are also concurrent; their point of concurrence is the isogonal conjugate \(Q\).
For example, \(\angle BAQ\) is the reflection of the angle between \(AP\) and \(AC\), hence corresponds to \(\angle CAP=\varphi\). Similarly, \(\angle ACQ\) corresponds to \(\angle BCP=\varphi\), and \(\angle CBQ\) corresponds to \(\angle ABP=\varphi\). Therefore \(\angle BAQ=\angle ACQ=\angle CBQ\), so \(Q\) is the second Brocard point.
The key is not to lose the cyclic order of equalities.