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We study the similarities between the Fano varieties of lines on a cubicfourfold, a hyper-Kaehler fourfold studied by Beauville and Donagi, and thehyper-Kaehler fourfold constructed by Debarre and Voisin. We exhibit an analogof the notion of "triangle" for these varieties and prove that the6-dimensional variety of "triangles" is a Lagrangian subvariety in the cube ofthe constructed hyper-Kaehler fourfold.

On the variety of triangles for a hyper-Kaehler fourfold constructed by Debarre and Voisin