{"title":"Two remarkable triangles of a triangle and their circumcircles","authors":"S. Kiss, Bálint Bíró","doi":"10.4171/EM/431","DOIUrl":"https://doi.org/10.4171/EM/431","url":null,"abstract":"","PeriodicalId":41994,"journal":{"name":"Elemente der Mathematik","volume":" ","pages":""},"PeriodicalIF":0.1,"publicationDate":"2021-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48723087","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Polynomial fill-in puzzles or how to make sense of the Cook–Levin theorem","authors":"A. Müller","doi":"10.4171/EM/439","DOIUrl":"https://doi.org/10.4171/EM/439","url":null,"abstract":"","PeriodicalId":41994,"journal":{"name":"Elemente der Mathematik","volume":" ","pages":""},"PeriodicalIF":0.1,"publicationDate":"2021-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48255083","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Sur un article d’Euler, prématuré bien que posthume","authors":"M. Ojanguren","doi":"10.4171/EM/438","DOIUrl":"https://doi.org/10.4171/EM/438","url":null,"abstract":"","PeriodicalId":41994,"journal":{"name":"Elemente der Mathematik","volume":" ","pages":""},"PeriodicalIF":0.1,"publicationDate":"2021-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46996949","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a convex n-gon P and a positive integer m such that 3 ≤ m ≤ n − 1, let Q denote the largest area convex m-gon contained in P . We are interested in the minimum value of ∆(Q)/∆(P ), the ratio of the areas of these two polygons. More precisely, given positive integers n and m, with 3 ≤ m ≤ n− 1, define fn(m) = min P∈Pn max Q⊂P,|Q|=m ∆(Q) ∆(P ) where the maximum is taken over all m-gons contained in P , and the minimum is taken over Pn, the entire class of convex n-gons. The values of f4(3), f5(4) and f6(3) are known. In this paper we compute the values of f5(3), f6(5) and f6(4). In addition, we prove that for all n ≥ 6 we have 4 n · sin (π n ) ≤ 1− fn(n− 1) ≤ min (
给定凸n边形P和正整数m，使得3≤m≤n−1，设Q表示P中包含的最大面积凸m边形。我们感兴趣的是∆（Q）/∆（P）的最小值，即这两个多边形的面积比。更准确地说，给定正整数n和m，其中3≤m≤n−1，定义fn（m）=min P∈Pn max Q⊂P，|Q|=m∆（Q）∆（P），其中最大值取P中包含的所有m边，最小值取Pn，整个凸n边类。f4（3）、f5（4）和f6（3）的值是已知的。在本文中，我们计算了f5（3）、f6（5）和f6（4）的值。此外，我们证明了对于所有n≥6，我们有4n·sin（πn）≤1−fn（n−1）≤min(
{"title":"On the maximum area of inscribed polygons","authors":"D. Ismailescu, Min Jung Kim, Eric Wang","doi":"10.4171/EM/442","DOIUrl":"https://doi.org/10.4171/EM/442","url":null,"abstract":"Given a convex n-gon P and a positive integer m such that 3 ≤ m ≤ n − 1, let Q denote the largest area convex m-gon contained in P . We are interested in the minimum value of ∆(Q)/∆(P ), the ratio of the areas of these two polygons. More precisely, given positive integers n and m, with 3 ≤ m ≤ n− 1, define fn(m) = min P∈Pn max Q⊂P,|Q|=m ∆(Q) ∆(P ) where the maximum is taken over all m-gons contained in P , and the minimum is taken over Pn, the entire class of convex n-gons. The values of f4(3), f5(4) and f6(3) are known. In this paper we compute the values of f5(3), f6(5) and f6(4). In addition, we prove that for all n ≥ 6 we have 4 n · sin (π n ) ≤ 1− fn(n− 1) ≤ min (","PeriodicalId":41994,"journal":{"name":"Elemente der Mathematik","volume":" ","pages":""},"PeriodicalIF":0.1,"publicationDate":"2021-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44231180","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Schröder’s processes and the best ways of increasing order of Newton’s method","authors":"F. Dubeau, C. Gnang","doi":"10.4171/EM/437","DOIUrl":"https://doi.org/10.4171/EM/437","url":null,"abstract":"","PeriodicalId":41994,"journal":{"name":"Elemente der Mathematik","volume":" ","pages":""},"PeriodicalIF":0.1,"publicationDate":"2021-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48151399","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Regular spatial hexagons","authors":"Fritz Siegerist, Karl Wirth","doi":"10.4171/EM/434","DOIUrl":"https://doi.org/10.4171/EM/434","url":null,"abstract":"","PeriodicalId":41994,"journal":{"name":"Elemente der Mathematik","volume":" ","pages":""},"PeriodicalIF":0.1,"publicationDate":"2021-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42107347","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Symmedian and isodynamic points relations","authors":"Paris Pamfilos","doi":"10.4171/EM/432","DOIUrl":"https://doi.org/10.4171/EM/432","url":null,"abstract":"","PeriodicalId":41994,"journal":{"name":"Elemente der Mathematik","volume":" ","pages":""},"PeriodicalIF":0.1,"publicationDate":"2021-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47864054","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A short synthetic proof of Thébault’s theorem","authors":"Đura Paunić","doi":"10.4171/EM/433","DOIUrl":"https://doi.org/10.4171/EM/433","url":null,"abstract":"","PeriodicalId":41994,"journal":{"name":"Elemente der Mathematik","volume":" ","pages":""},"PeriodicalIF":0.1,"publicationDate":"2021-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49234657","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A matrix viewpoint for various algebraic extensions","authors":"G. Abrams, P. N. Ánh","doi":"10.4171/EM/430","DOIUrl":"https://doi.org/10.4171/EM/430","url":null,"abstract":"","PeriodicalId":41994,"journal":{"name":"Elemente der Mathematik","volume":" ","pages":""},"PeriodicalIF":0.1,"publicationDate":"2021-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49286181","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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