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Parallelogram in triangle

psi20
in Applied Mathematics

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I've forgotten nearly everything in geometry, so I'm having trouble with this problem.

 

Imagine you have a triangle ABC. The midpoints of the line segments are D, E, and F. Prove that the quadrilateral ADFE is a parallelogram.

tripic3gt.jpg

 

AF:AC = AD:AB and FAD = BAC, so the triangles ABC and ADF are similar. Therefore the angles phi and theta are the same, so DF is parallel to BC.

 

Similarly, DE is parallel to AC (and therefore to AF) and EF is parallel to AB (and therefore to AD. As the opposite sides of our quadrilateral as parallel, it's therefore a parallelogram.

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