89.54 Another decreasing sequence of triangles
89.54 Another decreasing sequence of triangles
复制标题
89.54 另一个三角形递减序列
DOI：
10.1017/s0025557200177897
复制
发表时间：
2005
期刊：
影响因子：
--
通讯作者：
J. Scott
中科院分区：
文献类型：
--
作者：
J. Scott
89.54 另一个三角形递减序列 我们首先需要解释术语“踏板三角形”，它通常指由三角形 ABC 的高的脚形成的三角形，也可以用于由三角形 ABC 内点 P 的垂线的脚形成的三角形。 Neuberg 定理指出，源自内点 P 和主三角形 ABC 的嵌套踏板三角形序列中的每三个成员都是相似的 ([1])。通过重复使用循环四边形同一段中的等角定理，可以轻松证明这种三元相似性。也很容易证明，面积最大的踏板三角形是外心 O（内侧三角形）的踏板三角形，因此 Neuberg 序列中连续三角形的面积比不能超过 \，并且取决于主三角形和内部点 P 的选择。在这篇文章中，我们证明了通过使用三角形中线的长度作为其后继边获得的三角形序列具有二元相似性。
89.54 Another decreasing sequence of triangles We first need to explain that the term pedal triangle, which often refers to the triangle formed by the feet of the altitudes of a triangle ABC, may also be used for the triangle formed by the feet of the perpendiculars from a point P inside the triangle ABC. A theorem of Neuberg states that every third member of the sequence of nested pedal triangles derived from an interior point P and a host triangle ABC is similar ([1]). The proof for this ternary similarity follows easily by repeated use of the theorem for equal angles in the same segment of a cyclic quadrilateral. It is also easy to show that the pedal triangle of greatest area is that for the circumcentre O (the medial triangle) so that the area ratio of successive triangles in the Neuberg sequence cannot exceed \ and depends on both the choice of host triangle and point P within. In this note we show that the sequence of triangles obtained by using the lengths of the medians of a triangle as the sides of its successor has binary similarity.