| 词条 | Finsler–Hadwiger theorem |
| 释义 |
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 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.[1] StatementTo 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.[2] The square EFGH is called the Finsler–Hadwiger square of the two given squares.[3] ApplicationRepeated 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.[4] References1. ^{{citation | last1 = Finsler | first1 = P. | last2 = Hadwiger | first2 = H. | doi = 10.1007/BF01214300 | issue = 1 | journal = Commentarii Mathematici Helvetici | language = German | mr = 1509584 | pages = 316–326 | title = Einige Relationen im Dreieck | volume = 10 | year = 1937}}. See in particular p. 324. 2. ^{{citation | last1 = Alsina | first1 = Claudi | last2 = Nelsen | first2 = Roger B. | contribution = The Finsler–Hadwiger Theorem 8.5 | isbn = 9780883853481 | page = [https://books.google.com/books?id=mIT5-BN_L0oC&pg=PA125 125] | publisher = Mathematical Association of America | title = Charming Proofs: A Journey Into Elegant Mathematics | url = | year = 2010}}. 3. ^{{citation | last1 = Detemple | first1 = Duane | last2 = Harold | first2 = Sonia | doi = 10.1080/0025570X.1996.11996375 | issue = 1 | journal = Mathematics Magazine | jstor = 2691390 | mr = 1573131 | pages = 15–27 | title = A round-up of square problems | volume = 69 | year = 1996}}. See problem 8, pp. 20–21. 4. ^{{harvtxt|Detemple|Harold|1996}}, problem 15, pp. 25–26. External links{{commons category|Finsler-Hadwiger theorem}}
3 : Quadrilaterals|Euclidean geometry|Theorems in geometry |
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