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Visual geometry
Let your eyes be wrong firstWhat survives a shake

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.

Do not write an equation yet. Shake the figure first.

Diagonals & midpointsDifficulty · Medium
Hands-on scene

Drag one vertex and wreck the quadrilateral

Even when the outer shape becomes skewed or concave, its four midpoints rebuild a parallelogram.

PARALLELOGRAM ✓
Two opposite pairs stay parallel — the midpoint figure repairs itself

Play the reversal slowly

Now move the same pieces step by step and prove that the pattern you touched was not an accident.

Visual proof scrubber
Scene 1 / 4
PQRS

An arbitrary quadrilateral; mark the midpoints P, Q, R, S of its sides.

Return to the exact question

Show that joining the midpoints of the four sides of any quadrilateral yields a parallelogram.

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

Area = PQRS is a parallelogram (area ½)

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.

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