Proofs about General Polygons

The last problem was interesting.  We started with a lopsided quadrilateral and joined the midpoints of its sides to form another quadrilateral which turned out to be a parallelogram.  This suggests a Theorem, which we can prove using coordinates:

Theorem: 

If the midpoints of the sides of a quadrilateral are joined to form an inscribed quadrilateral, then that inscribed quadrilateral is a parallelogram.

Proof:

We can place one of the sides along the positive x-axis with its vertex, A, at the origin and label the coordinates of the other vertex, B, as (2b, 0), where b is some number (it is half the length of that side).  We can label the other vertices as C(2c, 2e) and D(2d, 2f) where c, e, d, and f are numbers, not necessarily all positive:

 

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Now we use the midpoint formula to find the midpoints of the sides:

Finally we use slope to prove sides MN and PQ are parallel:

and we can use slopes to prove sides NP and QM are also parallel:

So quadrilateral MNPQ is a parallelogram since its opposite sides are parallel.

 

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