![Van-Aubel-theorem combined](https://upload.wikimedia.org/wikipedia/commons/d/de/Van-Aubel-theorem_combined.svg)
In
plane geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
, Van Aubel's theorem describes a relationship between squares constructed on the sides of a
quadrilateral
In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
. Starting with a given convex quadrilateral, construct a
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
, 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 angles to one another. Another way of saying the same thing is that the center points of the four squares form the vertices of an
equidiagonal orthodiagonal quadrilateral
In Euclidean geometry, an orthodiagonal quadrilateral is a quadrilateral in which the diagonals cross at right angles. In other words, it is a four-sided figure in which the line segments between non-adjacent vertices are orthogonal (perpendicular ...
. 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 an academic journal of mathematics education, published three times yearly, that publishes "articles about the teaching and learning of mathematics with a focus on the 15–20 age range and expositions of attractive ...
''.
[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.
See also
*
Petr–Douglas–Neumann theorem
*
Thébault's theorem
Thébault's theorem is the name given variously to one of the geometry problems proposed by the French mathematician Victor Thébault, individually known as Thébault's problem I, II, and III.
Thébault's problem I
Given any parallelogram, co ...
*
Napoleon's theorem
In geometry, Napoleon's theorem states that if equilateral triangles are constructed on the sides of any triangle, either all outward or all inward, the lines connecting the centres of those equilateral triangles themselves form an equilateral tr ...
*
Napoleon points
In geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of ...
*
Bottema's theorem
References
External links
* {{MathWorld , urlname=vanAubelsTheorem , title=van Aubel's Theorem
Van Aubel's Theorem for Quadrilateralsan
Van Aubel's Theorem for Trianglesby Jay Warendorff,
The Wolfram Demonstrations Project
The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...
.
The Beautiful Geometric Theorem of Van Aubelby
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 appletby 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 Pointsby Chris Van Tienhoven a
Encyclopedia of Quadri-Figures (EQF)
Note concernant les centres de carrés construits sur les côtés d'un polygon quelconqueby M. H. Van Aubel a
HathiTrust Digital Library
Theorems about quadrilaterals