The Finsler–Hadwiger theorem is statement in

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

that describes a third square derived from any two

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 ...

s that share a vertex. The theorem is named after Paul Finsler and

Hugo Hadwiger Hugo Hadwiger (23 December 1908 in Karlsruhe, Germany – 29 October 1981 in Bern, Switzerland) was a Swiss people, Swiss mathematician, known for his work in geometry, combinatorics, and cryptography. Biography Although born in Karlsruhe, Ge ...

, who published it in 1937 as part of the same paper in which they published the

Hadwiger–Finsler inequality In mathematics, the Hadwiger–Finsler inequality is a result on the geometry of triangles in the Euclidean plane. It states that if a triangle in the plane has side lengths ''a'', ''b'' and ''c'' and area ''T'', then :a^ + b^ + c^ \geq (a - b)^ + ...

relating the side lengths and area of a triangle.

Statement

To state the theorem, suppose that ABCD and AB'C'D' are two squares with common vertex A. Let E and G be the midpoints of B'D and D'B respectively, and let F and H be the centers of the two squares. Then the theorem states that the quadrilateral EFGH is a square as well. The square EFGH is called the Finsler–Hadwiger square of the two given squares. It is the midsquare of the midsquare quadrilateral B'DBD'.

Application

Repeated application of the Finsler–Hadwiger theorem can be used to prove Van Aubel's theorem, on the congruence and perpendicularity of segments through centers of four squares constructed on the sides of an arbitrary quadrilateral. Each pair of consecutive squares forms an instance of the theorem, and the two pairs of opposite Finsler–Hadwiger squares of those instances form another two instances of the theorem, having the same derived square., problem 15, pp. 25–26.

References

External links

* {{DEFAULTSORT:Finsler-Hadwiger theorem Theorems about quadrilaterals Euclidean geometry