Choreographic Three Bodies on the Lemniscate: GIF animation

Three Bodies at t=K (black circles) and Center of Mass (red circle) for 0<=k^2<=1.

Horizontal componet of the Center of Mass g_{x} at t=K for 0<=k^2<=1.

At t=K, the Center of Mass is at the origin if and only if
k^2 = (2+sqrt{3})/4 = 0.93301270189221932338...

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