{"title":"具有公共面积和公共周长的积分三角形和垂直四边形对","authors":"A. S. Zargar, Yong Zhang","doi":"10.7169/facm/1842","DOIUrl":null,"url":null,"abstract":"By the theory of elliptic curves, we show that there are infinitely many integral right triangle-perpendicular quadrilateral, integral isosceles triangle-perpendicular quadrilateral, and Heron triangle-perpendicular quadrilateral pairs with a common area and a common perimeter. Moreover, for the elliptic curve associated to integral isosceles triangle and integral perpendicular quadrilateral pairs, we present several subfamilies of rank $\\geq 4$, and show the existence of infinitely many elliptic curves of rank $\\geq 5$, parameterized by the points of an elliptic curve of positive rank.","PeriodicalId":44655,"journal":{"name":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Integral triangles and perpendicular quadrilateral pairs with a common area and a common perimeter\",\"authors\":\"A. S. Zargar, Yong Zhang\",\"doi\":\"10.7169/facm/1842\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"By the theory of elliptic curves, we show that there are infinitely many integral right triangle-perpendicular quadrilateral, integral isosceles triangle-perpendicular quadrilateral, and Heron triangle-perpendicular quadrilateral pairs with a common area and a common perimeter. Moreover, for the elliptic curve associated to integral isosceles triangle and integral perpendicular quadrilateral pairs, we present several subfamilies of rank $\\\\geq 4$, and show the existence of infinitely many elliptic curves of rank $\\\\geq 5$, parameterized by the points of an elliptic curve of positive rank.\",\"PeriodicalId\":44655,\"journal\":{\"name\":\"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2020-12-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.7169/facm/1842\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"FUNCTIONES ET APPROXIMATIO COMMENTARII MATHEMATICI","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.7169/facm/1842","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Integral triangles and perpendicular quadrilateral pairs with a common area and a common perimeter
By the theory of elliptic curves, we show that there are infinitely many integral right triangle-perpendicular quadrilateral, integral isosceles triangle-perpendicular quadrilateral, and Heron triangle-perpendicular quadrilateral pairs with a common area and a common perimeter. Moreover, for the elliptic curve associated to integral isosceles triangle and integral perpendicular quadrilateral pairs, we present several subfamilies of rank $\geq 4$, and show the existence of infinitely many elliptic curves of rank $\geq 5$, parameterized by the points of an elliptic curve of positive rank.