plane geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...

, Van Aubel's theorem describes a relationship between squares constructed on the sides of a

quadrilateral In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''l ...

. Starting with a given convex quadrilateral, construct a

square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

, external to the quadrilateral, on each side. Van Aubel's theorem states that the two line segments between the centers of opposite squares are of equal lengths and are at

right angle In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...

s to one another. Another way of saying the same thing is that the center points of the four squares form the vertices of a

midsquare quadrilateral In elementary geometry, a quadrilateral whose diagonals are perpendicular and of equal length has been called a midsquare quadrilateral (referring to the square formed by its four edge midpoints). These shapes are, by definition, simultaneously ...

. The theorem is named after Belgian mathematician Henricus Hubertus (Henri) Van Aubel (1830–1906), who published it in 1878. The theorem holds true also for re-entrant quadrilaterals, Coxeter, H.S.M., and Greitzer, Samuel L. 1967. ''Geometry Revisited'', pages 52. and when the squares are constructed internally to the given quadrilateral.D. Pellegrinetti
"The Six-Point Circle for the Quadrangle"
''International Journal of Geometry'', Vol. 8 (Oct., 2019), No. 2, pp. 5–13. For complex (self-intersecting) quadrilaterals, the ''external'' and ''internal'' constructions for the squares are not definable. In this case, the theorem holds true when the constructions are carried out in the more general way: *follow the quadrilateral vertices in a sequential direction and construct each square on the right hand side of each side of the given quadrilateral. *Follow the quadrilateral vertices in the same sequential direction and construct each square on the left hand side of each side of the given quadrilateral. The segments joining the centers of the squares constructed externally (or internally) to the quadrilateral over two opposite sides have been referred to as ''Van Aubel segments''. The points of intersection of two equal and orthogonal Van Aubel segments (produced when necessary) have been referred to as ''Van Aubel points'': first or outer Van Aubel point for the external construction, second or inner Van Aubel point for the internal one. The Van Aubel theorem configuration presents some relevant features, among others: *the Van Aubel points are the centers of the two circumscribed squares of the quadrilateral.Ch. van Tienhoven, D. Pellegrinetti

''Journal for Geometry and Graphics'', Vol. 25 (July, 2021), No. 1, pp. 53–59. *The Van Aubel points, the mid-points of the quadrilateral diagonals and the mid-points of the Van Aubel segments are concyclic. A few extensions of the theorem, considering similar rectangles, similar rhombi and similar parallelograms constructed on the sides of the given quadrilateral, have been published on ''

The Mathematical Gazette ''The Mathematical Gazette'' is a triannual peer-reviewed academic journal published by Cambridge University Press on behalf of the Mathematical Association. It covers mathematics education with a focus on the 15–20 years age range. The journ ...

''.M. de Villiers
"Dual Generalizations of Van Aubel's theorem"
. ''The Mathematical Gazette'', Vol. 82 (Nov., 1998), pp. 405-412.J. R. Silvester
"Extensions of a Theorem of Van Aubel"
''The Mathematical Gazette'', Vol. 90 (Mar., 2006), pp. 2-12.

References

External links

* {{MathWorld , urlname=vanAubelsTheorem , title=van Aubel's Theorem
Van Aubel's Theorem for Quadrilaterals
an
Van Aubel's Theorem for Triangles
by Jay Warendorff,

The Wolfram Demonstrations Project The Wolfram Demonstrations Project is an open-source collection of interactive programmes called Demonstrations. It is hosted by Wolfram Research. At its launch, it contained 1300 demonstrations but has grown to over 10,000. The site won a Pa ...

.
The Beautiful Geometric Theorem of Van Aubel
by

Yutaka Nishiyama is a Japanese mathematician and professor at the Osaka University of Economics, where he teaches mathematics and information. He is known as the "boomerang professor". He has written nine books about the mathematics in daily life. The most recen ...

International Journal of Pure and Applied Mathematics

Interactive applet
by Tim Brzezinski showing Van Aubel's Theorem made usin
GeoGebra

at ttp://dynamicmathematicslearning.com/JavaGSPLinks.htm Dynamic Geometry Sketches interactive geometry sketches.
QG-2P6: Outer and Inner Van Aubel Points
by Chris Van Tienhoven a
Encyclopedia of Quadri-Figures (EQF)

Note concernant les centres de carrés construits sur les côtés d'un polygon quelconque
by M. H. Van Aubel a
HathiTrust Digital Library
Theorems about quadrilaterals

See also

References

External links