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Lemoine Circle and Isogonality

Prove that the First Lemoine Circle has Brocard midpoint (the center of Brocard circle) as a center.

See First Lemoine circle and Brocard circle.

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?

Note by Samuraiwarm Tsunayoshi
2 years, 7 months ago

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