The midpoints of the sides of a quadrilateral

Math Central

Quandaries & Queries

Question from Andrea, a student:

The segments, joining, in order the midpoints of consecutive sides of a quadrilateral form a parallelogram.

Hi Andrea,

I can help get you started.

In my diagram I labeled the vertices A, B, C and D, the midpoints of the sides E, F, G and H and drew the diagonal CA.

quadrilateral

Focus on the triangle ABC. If you can show CA is parallel to FE then you can use a similar argument to show A is parallel to GH.

triangle

In triangle ABC extend the line segment FE to P so that |FE|=|EP|. What can you say about triangles EBF and APE?

Penny

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