Let $ABCD$ be a parallelogram and $I,J$ are two points such that $AI = ID$ and $BJ = JD$. Show that ($IJ$) is parallel to ($AB$).
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I think that you meant that I is the middle of the line segment with end points A and D and J is the middle of the line segment with end points B and D, no? – user37238 Jun 30 '14 at 13:49
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I agree with @user37238. There have to be more constaints than you've given to be able to prove this. As is, any points $I$ and $J$ that make isoceles triangles $\bigtriangleup AID$ and $\bigtriangleup BJD$ will satisfy the conditions, and it's easy to show a counterexample to the statement you're trying to prove. – John Jun 30 '14 at 15:03