The Finsler–Hadwiger theorem is statement in Euclidean plane geometry that describes a third square derived from any two squares that share a vertex. The theorem is named after

Paul Finsler Paul Finsler (born 11 April 1894, in Heilbronn, Germany, died 29 April 1970 in Zurich, Switzerland) was a German and Swiss mathematician. Finsler did his undergraduate studies at the Technische Hochschule Stuttgart, and his graduate studies at ...

and Hugo Hadwiger, who published it in 1937 as part of the same paper in which they published the Hadwiger–Finsler inequality 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.

Application

Repeated application of the Finsler–Hadwiger theorem can be used to prove

Van Aubel's theorem In plane geometry, Van Aubel's theorem describes a relationship between squares constructed on the sides of a quadrilateral. Starting with a given convex quadrilateral, construct a square, external to the quadrilateral, on each side. Van Aubel's ...

, 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