由Brocard Point 發現幾何不等式

科展類別
臺灣國際科展
屆次
2006年
科別
數學科
學校名稱
臺北市立內湖高級中學
指導老師
劉紹正
作者
林祐民、張瀚之
關鍵字
法格乃諾問題 費馬點 尤拉公式

摘要或動機

本研究報告以Brocard Point 為核心,所用到的性質均先證明,以確認其正確性,並推演出一些其他的性質,藉由這些性質導出幾何不等式。內容可概分為四部份:(1)以Brocard Angle 及已知的或推演出的基本性質,導出一些不等式。(2)結合「法格乃諾問題」、「費馬點」、「尤拉公式」導出幾個幾何不等式。尤其是三角形邊長與面積,外接、內切圓半徑與邊長間的不等關係,頗為有趣。(3)以向量為工具,分別計算內、重、垂心與Brocard Point 間的距離,並導出邊長的不等關係。其中由內心及重心所導出的不等式,清楚俐落;垂心所導出的不等式則較為複雜。(4)以Brocard Cirle 與內、重心間的關係,導出一系列的不等式。其中Weitgenberk 不等式的無意發現,令我們印象深刻。The Discovery of Geometry Inequalities by Brocard Point This paper takes Brocard Point as a core. We proved some properties about Brocard geometry to confirm its accuracy, and deduce some other properties, and then derive some geometry inequalities by these properties. The content may divide into four parts: a) Derives geometry inequality by Brocard Angle, Crux Mathematicorum and properties which known or deduced. b) Unifies "Fagnano problem", "Fermat Point", "Euler formula" to derive several geometry inequalities. In particular the inequalities between triangle area and length of side, or circumradius inradius and the length of side, is quite interesting. c) Derives geometry inequalities about length of sides in triangle by the distances between incenter centroid circumcenter and Brocard Point. Especially, these inequalities were elegant which derived by incenter and centroid, but it was complicated derived by orthocenter. d) According to the relation about incenter centroid and Brocard Circle derives a series of inequalities. Discover Weitgenberk inequality makes us excited.


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