Four halfway points, one parallelogram
Show on a big screenJoin the midpoints of any four-sided shape, and a parallelogram appears.
- A lopsided quadrilateral ABCD.
- Mark the midpoint of each side: P, Q, R, S.
- Join P, Q, R and S in order.
- PQRS is a parallelogram, however lopsided ABCD looked.
The rule
Claim: Join the four side midpoints of any quadrilateral in order. They make a parallelogram.
P, Q, R, S are the midpoints of AB, BC, CD, DA. Join them in order: P, Q, R, S, then P again.
Join AC. It makes triangles ABC and ACD. In triangle ABC, PQ joins two midpoints. PQ is parallel to AC and half its length. In triangle ACD, SR does the same. So PQ and SR are equal and parallel. That makes PQRS a parallelogram.
| Corners of ABCD | Corners of PQRS |
|---|---|
| (0,0), (8,0), (10,6), (2,4) | (4,0), (9,3), (6,5), (1,2) |
Check with numbers. A pair like (8, 0) means 8 across and 0 up. Diagonal AC runs 10 across and 6 up. PQ runs 5 across and 3 up. SR runs 5 across and 3 up too. Each is half of AC, in the same direction.
The argument never used the shape of ABCD. Only the diagonal AC mattered. Non-convex — a shape with a corner pushed in. There, triangles ABC and ACD can overlap. The argument still holds. So PQRS is a parallelogram here too.
Special cases follow. PQ is half of AC. By the same steps, QR is half of BD. If the diagonals are equal, PQRS has equal sides. So it is a rhombus, with four equal sides. If the diagonals cross at right angles, PQRS has a right angle. So it is a rectangle.
Drag a corner. Then show the diagonals. PQ and SR are parallel to AC. Each is half of AC. QR and PS are parallel to BD. Each is half of BD.
Loading lab…
Try the lab to finish this lesson.