Problem
GEO-B3-M05-P023 Brocard Points on the Circle with Diameter \(OK\)
Let \(O\) be the circumcenter, \(K\) the Lemoine point, and \(P,Q\) the first and second Brocard points of triangle \(ABC\). Prove that \(P\) and \(Q\) lie on the circle with diameter \(OK\).
C. Hint 1. Consider three similar figures built on sides \(BC,CA,AB\).
D. Hint 2. Brocard points are centers of compatible spiral similarities of these figures.
Build three similarly oriented figures on sides \(BC,CA,AB\). Their similarity circle has diameter \(OK\): point \(O\) corresponds to the perpendicular bisectors of the sides, while point \(K\) corresponds to the lines through the Lemoine point parallel to the sides.
For the first Brocard point, the lines \(CP,AP,BP\) are corresponding lines of these three figures. Hence their common point \(P\) lies on the similarity circle. Similarly, the second Brocard point \(Q\) comes from the opposite orientation of corresponding lines and also lies on the same circle.
Since the similarity circle has diameter \(OK\), \(P\) and \(Q\) lie on the circle with diameter \(OK\).
A strong conceptual result; best after separate Lemoine and Brocard work.