Varignon’s Theorem — Midpoints of Any Quadrilateral Form a Parallelogram
Take any lopsided quadrilateral. Join the midpoints of its four sides and you always get a parallelogram — no matter how skewed.
Problem
Show that joining the midpoints of the four sides of any quadrilateral yields a parallelogram.
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An arbitrary quadrilateral; mark the midpoints P, Q, R, S of its sides.
Why this approach works
Draw diagonal AC. In triangle ABC, the segment PQ joining the midpoints of AB and BC is the midsegment — parallel to AC and half its length. Likewise SR (midpoints of CD, DA) is parallel to AC and half. So PQ ∥ SR and equal. Using diagonal BD gives QR ∥ PS. Two pairs of parallel equal sides — PQRS is a parallelogram, whatever the original shape. (Its area is exactly half the quadrilateral.)
Proof
PQ and SR are both parallel to diagonal AC and half its length, so PQ∥SR; BD gives QR∥PS. Two pairs of parallel sides make PQRS a parallelogram, independent of the original shape.
Try it yourself
Drag a vertex anywhere (even concave): the midpoint quadrilateral stays a parallelogram.
Related mathematics
Beyond MathVoyage
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