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Consider the set of lines that separate a triangle in two parts of same area. The three median belong to the set, in particular. What can be said of the envelope of the set of lines? For example, is the envelope a bounded set?

Thank you for your help.

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    @joriki: I see.2011-08-05

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See this Wikipedia section. The envelope is a deltoid with vertices at the midpoints of the medians.

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    The linked Wiki article correctly points out that the envelope consists of three pieces of hyperbolas. The curve that is sometimes called "the" deltoid" in books on curves is this curve: http://en.wikipedia.org/wiki/Deltoid_curve2011-08-05