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?

#Geometry

Easy Math Editor

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$