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Proving with deductive geometry

Hi, I need some help in solving this question:

"Draw any quadrilateral, find the midpoints of its sides and join consecutive midpoints to form an inner quadrilateral. Make a hypothesis about the inner quad. Use deductive geometry to prove your hypothesis is true."

This is what I got so far:

Attachment 21883

The quadrilateral inside seems to be always a parallelogram no matter what shape the outside quadrilateral is like.

The hard part is that i don't how to use deductive geometry to prove this. (Doh) (I do know some deductive geometry, but I just don't know where to start.)

Re: Proving with deductive geometry

The proof depends heavily upon the axioms and definitions you have to use. We do not know those. So the best we can do here is to give a outline.

If you connect two non-consecutive vertices of the original quadrilateral you have a diagonal. There are two sub-triangles formed. Now the segment formed by joining the midpoints of the two adjacent sides is parallel to and one half the length of the third side, that diagonal. Do the same on the second triangle. Now you have a quadrilateral have opposite sides parallel of the same length.