Varignon set

The set of Varignon (also set by the center square in the) describes geometry a property of Four Corners . It is named after Pierre de Varignon (1654–1722).

formulation

If you connect the centers of neighboring sides of a square, you get a parallelogram .

proof

requirement

${\displaystyle {\overline {AE}}={\overline {EB}},{\overline {BF}}={\overline {FC}},{\overline {CG}}={\overline {GD}},{\overline {DH}}={\overline {HA}}}$

claim

The square EFGH is a parallelogram.

Course of evidence

1. Look at the triangle ABC. If you take B as the center of stretching of a centric stretch , A is mapped to E and C to F with a stretch factor ½. According to the imaging properties of centric stretching - the image is straight and the original line are parallel - it follows that AC ∥ EF.
2. Likewise one shows that AC ∥ GH, BD ∥ FG, and BD∥ HE.
3. The parallelism is transitive . So EF ∥ HG and FG ∥ HE.

The opposite sides of the quadrangle EFGH are parallel, which corresponds to the definition of a parallelogram.

Inferences

Circumference of the Varignon parallelogram

The circumference of the Varignon parallelogram is exactly as large as the sum of the diagonals in the square of the origin.

Area of ​​the Varignon parallelogram

The area of ​​the Varignon parallelogram is half as large as the area of ​​the square of origin.

Trivia

The so-called Varignon apparatus is a profane application of the mathematical theorems and can be used to optimize the location. Several locations are drawn to scale on a table top. At these locations, holes are drilled through which threads are pulled. The ends of all threads are knotted together on the top of the table. The corresponding weights involved are hung on the threads below the table top. For example, the number of people or the number of residents is used as the weight to express the weighting of the location. The forces that are now acting pull the node on the surface of the plate to the optimal location.

See also

• Commandino's theorem
• Diagonal set

literature

• Siegfried Krauter, Christine Bescherer: The Elementary Geometry Experience: A workbook for independent and active discovery . Springer, 2012, ISBN 978-3-8274-3025-0 , pp. 76-77
• HSM Coxeter, SL Greitzer: Geometry Revisited . MAA, Washington 1967, pp. 52-54
• Peter N. Oliver: Pierre Varignon and the Parallelogram Theorem (PDF) In: Mathematics Teacher , Volume 94, No. 4, April 2001, pp. 316-319
• Peter N. Oliver: Consequences of Varignon Parallelogram Theorem (PDF) In: Mathematics Teacher , Volume 94, No. 5, May 2001, pp. 406-408

Web links

• Eric W. Weisstein : Varignon's Theorem . In: MathWorld (English).
• Varignon-Parallelogram in Compendium Geometry (English)
• Varignon's theorem at Matroids Matheplanet
• Varignon set with Java animations
• Varignon parallelogram on cut-the-knot-org

Individual evidence

1. ↑ Math Prism. Retrieved August 17, 2020 .