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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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Do you mean lines or line segments? The set of line segements is bounded by the triangles. A single line is unbounded. –  Ross Millikan Aug 5 '11 at 16:10
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I mean lines. I wonder if the envelope of all the lines is bounded –  galath Aug 5 '11 at 16:13
    
Iin the example the envelope continues to infinity close to the x and y axes-it doesn't stop at 10. If you include a line, your envelope will be unbounded. –  Ross Millikan Aug 5 '11 at 16:29
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@Ross: That's not true. In this case the lines turn by $\pi$ and their envelope traces a bounded curve while they do. The unboundedness of the lines says nothing about the boundedness of the envelope. –  joriki Aug 5 '11 at 16:33
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@joriki: I see. –  Ross Millikan Aug 5 '11 at 16:35

1 Answer 1

up vote 3 down vote accepted

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: en.wikipedia.org/wiki/Deltoid_curve –  Joseph Malkevitch Aug 5 '11 at 19:55

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