[{"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","day":"28","year":"2018","date_published":"2018-09-28T00:00:00Z","publisher":"Taylor & Francis","article_processing_charge":"No","_id":"19494","author":[{"full_name":"Chan, Yik Tung","last_name":"Chan","first_name":"Yik Tung","id":"c4c0afc8-9262-11ed-9231-d8b0bc743af1","orcid":"0000-0001-8467-4106"}],"status":"public","volume":125,"arxiv":1,"publication_identifier":{"eissn":["1930-0972"],"issn":["0002-9890"]},"issue":"8","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.1706.05919","open_access":"1"}],"citation":{"mla":"Chan, Stephanie. “Rational Right Triangles of a given Area.” <i>The American Mathematical Monthly</i>, vol. 125, no. 8, Taylor &#38; Francis, 2018, pp. 689–703, doi:<a href=\"https://doi.org/10.1080/00029890.2018.1495491\">10.1080/00029890.2018.1495491</a>.","ama":"Chan S. Rational right triangles of a given area. <i>The American Mathematical Monthly</i>. 2018;125(8):689-703. doi:<a href=\"https://doi.org/10.1080/00029890.2018.1495491\">10.1080/00029890.2018.1495491</a>","ista":"Chan S. 2018. Rational right triangles of a given area. The American Mathematical Monthly. 125(8), 689–703.","apa":"Chan, S. (2018). Rational right triangles of a given area. <i>The American Mathematical Monthly</i>. Taylor &#38; Francis. <a href=\"https://doi.org/10.1080/00029890.2018.1495491\">https://doi.org/10.1080/00029890.2018.1495491</a>","ieee":"S. Chan, “Rational right triangles of a given area,” <i>The American Mathematical Monthly</i>, vol. 125, no. 8. Taylor &#38; Francis, pp. 689–703, 2018.","short":"S. Chan, The American Mathematical Monthly 125 (2018) 689–703.","chicago":"Chan, Stephanie. “Rational Right Triangles of a given Area.” <i>The American Mathematical Monthly</i>. Taylor &#38; Francis, 2018. <a href=\"https://doi.org/10.1080/00029890.2018.1495491\">https://doi.org/10.1080/00029890.2018.1495491</a>."},"article_type":"original","OA_type":"green","type":"journal_article","quality_controlled":"1","oa_version":"Preprint","language":[{"iso":"eng"}],"scopus_import":"1","publication_status":"published","date_updated":"2025-07-10T11:51:49Z","abstract":[{"text":"Starting from any given rational-sided, right triangle, for example, the (3,4,5)-triangle with area 6, we use Euclidean geometry to show that there are infinitely many other rational-sided, right triangles of the same area. We show further that the set of all such triangles of a given area is finitely generated under our geometric construction. Such areas are known as “congruent numbers” and have a rich history in which all the results in this article have been proved and far more. Yet, as far as we can tell, this seems to be the first exploration using this kind of geometric technique.","lang":"eng"}],"title":"Rational right triangles of a given area","external_id":{"arxiv":["1706.05919"]},"doi":"10.1080/00029890.2018.1495491","OA_place":"repository","page":"689-703","extern":"1","publication":"The American Mathematical Monthly","date_created":"2025-04-05T10:51:16Z","month":"09","intvolume":"       125","oa":1},{"intvolume":"       124","oa":1,"date_created":"2018-12-11T11:49:09Z","page":"588 - 596","publication":"The American Mathematical Monthly","month":"01","isi":1,"title":"On the lengths of curves passing through boundary points of a planar convex shape","abstract":[{"lang":"eng","text":"We study the lengths of curves passing through a fixed number of points on the boundary of a convex shape in the plane. We show that, for any convex shape K, there exist four points on the boundary of K such that the length of any curve passing through these points is at least half of the perimeter of K. It is also shown that the same statement does not remain valid with the additional constraint that the points are extreme points of K. Moreover, the factor &amp;#xbd; cannot be achieved with any fixed number of extreme points. We conclude the paper with a few other inequalities related to the perimeter of a convex shape."}],"external_id":{"isi":["000413947300002"],"arxiv":["1605.07997"]},"doi":"10.4169/amer.math.monthly.124.7.588","language":[{"iso":"eng"}],"oa_version":"Submitted Version","scopus_import":"1","publication_status":"published","date_updated":"2025-07-10T12:01:35Z","citation":{"mla":"Akopyan, Arseniy, and Vladislav Vysotsky. “On the Lengths of Curves Passing through Boundary Points of a Planar Convex Shape.” <i>The American Mathematical Monthly</i>, vol. 124, no. 7, Mathematical Association of America, 2017, pp. 588–96, doi:<a href=\"https://doi.org/10.4169/amer.math.monthly.124.7.588\">10.4169/amer.math.monthly.124.7.588</a>.","ieee":"A. Akopyan and V. Vysotsky, “On the lengths of curves passing through boundary points of a planar convex shape,” <i>The American Mathematical Monthly</i>, vol. 124, no. 7. Mathematical Association of America, pp. 588–596, 2017.","apa":"Akopyan, A., &#38; Vysotsky, V. (2017). On the lengths of curves passing through boundary points of a planar convex shape. <i>The American Mathematical Monthly</i>. Mathematical Association of America. <a href=\"https://doi.org/10.4169/amer.math.monthly.124.7.588\">https://doi.org/10.4169/amer.math.monthly.124.7.588</a>","chicago":"Akopyan, Arseniy, and Vladislav Vysotsky. “On the Lengths of Curves Passing through Boundary Points of a Planar Convex Shape.” <i>The American Mathematical Monthly</i>. Mathematical Association of America, 2017. <a href=\"https://doi.org/10.4169/amer.math.monthly.124.7.588\">https://doi.org/10.4169/amer.math.monthly.124.7.588</a>.","short":"A. Akopyan, V. Vysotsky, The American Mathematical Monthly 124 (2017) 588–596.","ama":"Akopyan A, Vysotsky V. On the lengths of curves passing through boundary points of a planar convex shape. <i>The American Mathematical Monthly</i>. 2017;124(7):588-596. doi:<a href=\"https://doi.org/10.4169/amer.math.monthly.124.7.588\">10.4169/amer.math.monthly.124.7.588</a>","ista":"Akopyan A, Vysotsky V. 2017. On the lengths of curves passing through boundary points of a planar convex shape. The American Mathematical Monthly. 124(7), 588–596."},"article_type":"original","project":[{"call_identifier":"FP7","name":"International IST Postdoc Fellowship Programme","_id":"25681D80-B435-11E9-9278-68D0E5697425","grant_number":"291734"}],"quality_controlled":"1","type":"journal_article","publication_identifier":{"issn":["0002-9890"]},"issue":"7","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1605.07997"}],"ec_funded":1,"status":"public","volume":124,"arxiv":1,"publist_id":"6534","department":[{"_id":"HeEd"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_published":"2017-01-01T00:00:00Z","year":"2017","day":"01","publisher":"Mathematical Association of America","_id":"909","author":[{"orcid":"0000-0002-2548-617X","first_name":"Arseniy","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","last_name":"Akopyan","full_name":"Akopyan, Arseniy"},{"last_name":"Vysotsky","full_name":"Vysotsky, Vladislav","first_name":"Vladislav"}],"article_processing_charge":"No"},{"month":"01","page":"614 - 618","publication":"American Mathematical Monthly","extern":"1","date_created":"2018-12-11T12:06:49Z","intvolume":"        96","date_updated":"2022-02-11T12:59:01Z","publication_status":"published","oa_version":"None","language":[{"iso":"eng"}],"doi":"10.1080/00029890.1989.11972250","title":"On the number of furthest neighbor pairs in a point set","issue":"7","main_file_link":[{"url":"http://www.jstor.org/stable/2325182 "}],"publication_identifier":{"eissn":["1930-0972"],"issn":["0002-9890"]},"type":"journal_article","quality_controlled":"1","article_type":"original","citation":{"chicago":"Edelsbrunner, Herbert, and Steven Skiena. “On the Number of Furthest Neighbor Pairs in a Point Set.” <i>American Mathematical Monthly</i>. Mathematical Association of America, 1989. <a href=\"https://doi.org/10.1080/00029890.1989.11972250\">https://doi.org/10.1080/00029890.1989.11972250</a>.","short":"H. Edelsbrunner, S. Skiena, American Mathematical Monthly 96 (1989) 614–618.","apa":"Edelsbrunner, H., &#38; Skiena, S. (1989). On the number of furthest neighbor pairs in a point set. <i>American Mathematical Monthly</i>. Mathematical Association of America. <a href=\"https://doi.org/10.1080/00029890.1989.11972250\">https://doi.org/10.1080/00029890.1989.11972250</a>","ieee":"H. Edelsbrunner and S. Skiena, “On the number of furthest neighbor pairs in a point set,” <i>American Mathematical Monthly</i>, vol. 96, no. 7. Mathematical Association of America, pp. 614–618, 1989.","ama":"Edelsbrunner H, Skiena S. On the number of furthest neighbor pairs in a point set. <i>American Mathematical Monthly</i>. 1989;96(7):614-618. doi:<a href=\"https://doi.org/10.1080/00029890.1989.11972250\">10.1080/00029890.1989.11972250</a>","ista":"Edelsbrunner H, Skiena S. 1989. On the number of furthest neighbor pairs in a point set. American Mathematical Monthly. 96(7), 614–618.","mla":"Edelsbrunner, Herbert, and Steven Skiena. “On the Number of Furthest Neighbor Pairs in a Point Set.” <i>American Mathematical Monthly</i>, vol. 96, no. 7, Mathematical Association of America, 1989, pp. 614–18, doi:<a href=\"https://doi.org/10.1080/00029890.1989.11972250\">10.1080/00029890.1989.11972250</a>."},"_id":"4079","article_processing_charge":"No","author":[{"first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner"},{"full_name":"Skiena, Steven","last_name":"Skiena","first_name":"Steven"}],"publisher":"Mathematical Association of America","day":"01","year":"1989","date_published":"1989-01-01T00:00:00Z","user_id":"ea97e931-d5af-11eb-85d4-e6957dddbf17","publist_id":"2042","acknowledgement":"Research supported by Amoco Fnd. Fac. Dev. Comput. Sci. 1-6-44862","volume":96,"status":"public"}]