Prove that the First Lemoine Circle has Brocard midpoint (the center of Brocard circle) as a center.
My first wild guess was that First Lemoine Circle has centroid as a center, because Lemoine point is isogonal conjugate to centroid. But it turns out wrong. Why?!
From the theorem of Second Lemoine Circle we know that Second Lemoine Circle has a center exactly at Lemoine Point. Isogonality, the center of First Lemoine Circle should be at centroid, isn't it?