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In geometry, the midpoint polygon of a polygon $P$ is the polygon whose vertices are the midpoints of the edges of $P$ .^[1]^[2] It is sometimes called the Kasner polygon after Edward Kasner, who termed it the inscribed polygon "for brevity".^[3]^[4]

Examples

Triangle

The midpoint polygon of a triangle is called the medial triangle. It shares the same centroid and medians with the original triangle. The perimeter of the medial triangle equals the semiperimeter of the original triangle, and the area is one quarter of the area of the original triangle. This can be proven by the midpoint theorem of triangles and Heron's formula. The orthocenter of the medial triangle coincides with the circumcenter of the original triangle.

Quadrilateral

The midpoint polygon of a quadrilateral is a parallelogram called its Varignon parallelogram. If the quadrilateral is simple, the area of the parallelogram is one half the area of the original quadrilateral. The perimeter of the parallelogram equals the sum of the diagonals of the original quadrilateral.

References

^ Gardner 2006, p. 36.
^ Gardner & Gritzmann 1999, p. 92.
^ Kasner 1903, p. 59.
^ Schoenberg 1982, pp. 91, 101.

Gardner, Richard J. (2006), Geometric tomography, Encyclopedia of Mathematics and its Applications, vol. 58 (2nd ed.), Cambridge University Press
Gardner, Richard J.; Gritzmann, Peter (1999), "Uniqueness and Complexity in Discrete Tomography", in Herman, Gabor T.; Kuba, Attila (eds.), Discrete tomography: Foundations, Algorithms, and Applications, Springer, pp. 85–114
Kasner, Edward (March 1903), "The Group Generated by Central Symmetries, with Application to Polygons", American Mathematical Monthly, 10 (3): 57–63, doi:10.2307/2968300, JSTOR 2968300
Schoenberg, I. J. (1982), Mathematical time exposures, Mathematical Association of America, ISBN 0-88385-438-4