词条 | Heronian triangle | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
释义 |
In geometry, a Heronian triangle is a triangle that has side lengths and area that are all integers.[1][2] Heronian triangles are named after Hero of Alexandria. The term is sometimes applied more widely to triangles whose sides and area are all rational numbers,[3] since one can rescale the sides by a common multiple to obtain a triangle that is Heronian in the above sense. PropertiesAny right-angled triangle whose sidelengths are a Pythagorean triple is a Heronian triangle, as the side lengths of such a triangle are integers, and its area is also an integer, being half of the product of the two shorter sides of the triangle, at least one of which must be even. An example of a Heronian triangle which is not right-angled is the isosceles triangle with sidelengths 5, 5, and 6, whose area is 12. This triangle is obtained by joining two copies of the right-angled triangle with sides 3, 4, and 5 along the sides of length 4. This approach works in general, as illustrated in the adjacent picture. One takes a Pythagorean triple (a, b, c), with c being largest, then another one (a, d, e), with e being largest, constructs the triangles with these sidelengths, and joins them together along the sides of length a, to obtain a triangle with integer side lengths c, e, and b + d, and with area (one half times the base times the height). If a is even then the area A is an integer. Less obviously, if a is odd then A is still an integer, as b and d must both be even, making b+d even too. Some Heronian triangles cannot be obtained by joining together two right-angled triangles with integer sides as described above. For example, a 5, 29, 30 Heronian triangle with area 72 cannot be constructed from two integer Pythagorean triangles since none of its altitudes are integers. Also no primitive Pythagorean triangle can be constructed from two smaller integer Pythagorean triangles.[4]{{rp|p.17}} Such Heronian triangles are known as indecomposable.[4] However, if one allows Pythagorean triples with rational values, not necessarily integers, then a decomposition into right triangles with rational sides always exists,[5] because every altitude of a Heronian triangle is rational (since it equals twice the integer area divided by the integer base). So the Heronian triangle with sides 5, 29, 30 can be constructed from rational Pythagorean triangles with sides 7/5, 24/5, 5 and 143/5, 24/5, 29. Note that a Pythagorean triple with rational values is just a scaled version of a triple with integer values. Other properties of Heronian triangles are as follows:
Exact formula for all Heronian trianglesThe Indian mathematician Brahmagupta (598-668 A.D.) derived the parametric solution such that every Heronian triangle has sides proportional to:[18][19] for integers m, n and k where: . The proportionality factor is generally a rational {{frac|p|q}} where q = gcd(a, b, c) reduces the generated Heronian triangle to its primitive and p scales up this primitive to the required size. For example, taking m = 36, n = 4 and k = 3 produces a triangle with a = 5220, b = 900 and c = 5400, which is similar to the 5, 29, 30 Heronian triangle and the proportionality factor used has p = 1 and q = 180. The obstacle for a computational use of Brahmagupta’s parametric solution is the denominator q of the proportionality factor. q can only be determined by calculating the greatest common divisor of the three sides ( gcd(a, b, c) ) and introduces an element of unpredictability into the generation process.[19] The easiest way of generating lists of Heronian triangles is to generate all integer triangles up to a maximum side length and test for an integral area. Faster algorithms have been derived by {{harvtxt|Kurz|2008}}. There are infinitely many primitive and indecomposable non-Pythagorean Heronian triangles with integer values for the inradius and all three of the exradii, including the ones generated by[20]{{rp|Thm. 4}} There are infinitely many Heronian triangles that can be placed on a lattice such that not only are the vertices at lattice points, as holds for all Heronian triangles, but additionally the centers of the incircle and excircles are at lattice points.[20]{{rp|Thm. 5}} See also formulas for Heronian triangles with one angle equal to twice another, Heronian triangles with sides in arithmetic progression, and isosceles Heronian triangles. ExamplesThe list of primitive integer Heronian triangles, sorted by area and, if this is the same, by perimeter, starts as in the following table. "Primitive" means that the greatest common divisor of the three side lengths equals 1.
Lists of primitive Heronian triangles whose sides do not exceed 6,000,000 can be found at {{cite web |url=http://www.wm.uni-bayreuth.de/index.php?id=554&L=3|title=Lists of primitive Heronian triangles|publisher=Sascha Kurz, University of Bayreuth, Germany|access-date=29 March 2016}} Equable trianglesA shape is called equable if its area equals its perimeter. There are exactly five equable Heronian triangles: the ones with side lengths (5,12,13), (6,8,10), (6,25,29), (7,15,20), and (9,10,17).[21][22] Almost-equilateral Heronian trianglesSince the area of an equilateral triangle with rational sides is an irrational number, no equilateral triangle is Heronian. However, there is a unique sequence of Heronian triangles that are "almost equilateral" because the three sides are of the form n − 1, n, n + 1. A method for generating all solutions to this problem based on continued fractions was described in 1864 by Edward Sang,[23] and in 1880 Reinhold Hoppe gave a closed-form expression for the solutions.[24] The first few examples of these almost-equilateral triangles are listed in the following table {{OEIS|A003500}}:
Subsequent values of n can be found by multiplying the previous value by 4, then subtracting the value prior to that one (52 = 4 × 14 − 4, 194 = 4 × 52 − 14, etc.), thus: where t denotes any row in the table. This is a Lucas sequence. Alternatively, the formula generates all n. Equivalently, let A = area and y = inradius, then, where {n, y} are solutions to n2 − 12y2 = 4. A small transformation n = 2x yields a conventional Pell equation x2 − 3y2 = 1, the solutions of which can then be derived from the regular continued fraction expansion for {{radic|3}}.[25] The variable n is of the form , where k is 7, 97, 1351, 18817, …. The numbers in this sequence have the property that k consecutive integers have integral standard deviation.[26] See also
References1. ^{{citation |first1=John R. |last1=Carlson |title=Determination of Heronian Triangles |journal=Fibonacci Quarterly |volume=8 |pages=499–506 |year=1970 |url=http://www.fq.math.ca/Scanned/8-5/carlson-a.pdf }} 2. ^{{citation |first1=Raymond A. |last1=Beauregard |first2=E. R. |last2=Suryanarayan |title=The Brahmagupta Triangles |journal=College Mathematics Journal |volume=29 |issue=1 |pages=13–17 |date=January 1998 |doi= 10.2307/2687630|url=http://www.maa.org/mathdl/CMJ/methodoflastresort.pdf |jstor=2687630 }} 3. ^{{MathWorld |title=Heronian Triangle |id=HeronianTriangle}} 4. ^1 2 {{citation |first=Paul |last=Yiu |title=Heron triangles which cannot be decomposed into two integer right triangles |url=http://math.fau.edu/yiu/Southern080216.pdf |year=2008 |publisher=41st Meeting of Florida Section of Mathematical Association of America }} 5. ^{{citation |authorlink=Wacław Sierpiński |first=Wacław |last=Sierpiński |title=Pythagorean Triangles |url=https://books.google.com/books?id=6vOfpjmCd7sC |year=2003 |publisher=Dover Publications, Inc. |isbn=978-0-486-43278-6 |origyear=1962}} 6. ^1 {{Cite journal |title=On Heron Simplices and Integer Embedding |last=Friche |first=Jan |publisher=Ernst-Moritz-Arndt Universät Greiswald Publication |date=2 January 2002 |arxiv=math/0112239 }} 7. ^{{cite journal |last=Buchholz |first=R. H. |last2=MacDougall |first2=J. A. |title=Cyclic Polygons with Rational Sides and Area |citeseerx = 10.1.1.169.6336 |year=2001 |page=3 |publisher=CiteSeerX Penn State University }} 8. ^1 {{cite web|last=Somos|first=M.|authorlink=Michael Somos|title=Rational triangles|url=http://grail.eecs.csuohio.edu/~somos/rattri.html|date=December 2014|accessdate=2018-11-04}} 9. ^Mitchell, Douglas W. (2013), "Perpendicular Bisectors of Triangle Sides", Forum Geometricorum 13, 53−59: Theorem 2. 10. ^1 {{Cite journal |title=Determination of Heronian triangles |last=Carlson |first=John R. |publisher=San Diego State College |year=1970 |url=http://www.fq.math.ca/Scanned/8-5/carlson-a.pdf}} 11. ^{{Cite journal |title=Heron Quadrilaterals with sides in Arithmetic or Geometric progression |last=Buchholz |first=R. H. |last2=MacDougall |first2=J. A. |journal=Bulletin of the Australian Mathematical Society |pages=263–269 |volume=59 |year=1999 |url=http://journals.cambridge.org/article_S0004972700032883}} 12. ^{{Cite journal |title=On Triangles with Rational Sides and Having Rational Areas |last=Blichfeldt |first=H. F. |journal=Annals of Mathematics |volume=11 |issue=1/6 |year=1896–1897 |pages=57–60 |jstor=1967214 |doi=10.2307/1967214 }} 13. ^[https://arxiv.org/ftp/arxiv/papers/0803/0803.3778.pdf Zelator, K., "Triangle Angles and Sides in Progression and the diophantine equation x2+3y2=z2", Cornell Univ. archive, 2008] 14. ^Bailey, Herbert, and DeTemple, Duane, "Squares inscribed in angles and triangles", Mathematics Magazine 71(4), 1998, 278–284. 15. ^Clark Kimberling, "Trilinear distance inequalities for the symmedian point, the centroid, and other triangle centers", Forum Geometricorum, 10 (2010), 135−139. http://forumgeom.fau.edu/FG2010volume10/FG201015index.html 16. ^Clark Kimberling's Encyclopedia of Triangle Centers {{cite web|url=http://faculty.evansville.edu/ck6/encyclopedia/ETC.html |title=Encyclopedia of Triangle Centers |accessdate=2012-06-17 |deadurl=yes |archiveurl=https://web.archive.org/web/20120419171900/http://faculty.evansville.edu/ck6/encyclopedia/ETC.html |archivedate=2012-04-19 |df= }} 17. ^Yiu, P., "Heronian triangles are lattice triangles", American Mathematical Monthly 108 (2001), 261–263. 18. ^Carmichael, R. D., 1914, "Diophantine Analysis", pp.11-13; in R. D. Carmichael, 1959, The Theory of Numbers and Diophantine Analysis, Dover Publications, Inc. 19. ^1 {{cite journal | last = Kurz | first = Sascha | arxiv = 1401.6150 | issue = 2 | journal = Serdica Journal of Computing | mr = 2473583 | pages = 181–196 | title = On the generation of Heronian triangles | url = https://eudml.org/doc/11461 | volume = 2 | year = 2008 | ref = harv| bibcode = 2014arXiv1401.6150K }}. 20. ^1 Zhou, Li, "Primitive Heronian Triangles With Integer Inradius and Exradii", Forum Geometricorum 18, 2018, 71-77. http://forumgeom.fau.edu/FG2018volume18/FG201811.pdf 21. ^{{citation|title=History of the Theory of Numbers, Volume Il: Diophantine Analysis|first=Leonard Eugene|last=Dickson|authorlink=L. E. Dickson|publisher=Dover Publications|year=2005|isbn=9780486442334|page=199|title-link=History of the Theory of Numbers}} 22. ^{{citation | last = Markowitz | first = L. | issue = 3 | journal = The Mathematics Teacher | pages = 222–3 | title = Area = Perimeter | volume = 74 | year = 1981}} 23. ^{{citation|title=On the theory of commensurables|first=Edward|last=Sang|authorlink=Edward Sang|journal=Transactions of the Royal Society of Edinburgh|volume=23|pages=721–760}}. See in particular [https://books.google.com/books?id=uXAxAQAAMAAJ&pg=PA734 p. 734]. 24. ^{{citation|title=A triangle with integral sides and area|first=H. W.|last=Gould|journal=Fibonacci Quarterly|pages=27–39|date=February 1973|volume=11|issue=1|url=http://www.mathstat.dal.ca/FQ/Scanned/11-1/gould.pdf}}. 25. ^{{citation |first1=William H. |last1=Richardson |title=Super-Heronian Triangles |date=2007 |url=http://www.math.wichita.edu/~richardson/heronian/heronian.html}} 26. ^Online Encyclopedia of Integer Sequences, {{OEIS2C|A011943}}. External links
3 : Arithmetic problems of plane geometry|Triangles|Articles containing proofs |
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