Dynamic Triangle Geometry: Families of Lines with Equal Intercepts
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namic Triangle Geometry: Families of Lines with Equal Intercepts Paul Yiu
Published online: 10 July 2008 Ó Springer Science+Business Media B.V. 2008
1 Preliminaries This snapshot outlines an experiment in triangle geometry using the Geometer’s Sketchpadr which enables students to explore and discover interesting results which are expressible in elementary terms in Advanced Euclidean Geometry.1 We shall assume basic commands of the Geometer’s Sketchpadr and the availability of a tool folder containing the following tools2 for the classical centers of a triangle: (1) (2) (3)
the centroid (G in Fig. 1), the circumcenter (O in Fig. 1) and circumcircle, the orthocenter (H in Fig. 1),
1
The contents of this paper form part of a course on Advanced Euclidean Geometry which the author periodically teaches to prospective and in-service high school and community college teachers (see Yiu 2001). The students are first introduced to the most basic notions of Advanced Euclidean Geometry, some basic techniques of ruler-and-compass constructions performed by the Geometer’s Sketchpadr ; and then to the use of homogeneous barycentric coordinates in triangle geometry. The course emphasizes the use of the Geometer’s Sketchpadr as an exploratory tool, and the testing of results of Geometer’s Sketchpadr explorations by computations with homogeneous barycentric coordinates. 2
A construction appearing in san serif is assumed to be readily performed with a customized tool, or a builtin command, in Geometer’s Sketchpadr :
* This column will publish short (from just a few paragraphs to ten or so pages), lively and intriguing computer-related mathematics vignettes. These vignettes or snapshots should illustrate ways in which computer environments have transformed the practice of mathematics or mathematics pedagogy. They could also include puzzles or brain-teasers involving the use of computers or computational theory. Snapshots are subject to peer review. From the Column Editor Uri Wilensky, Northwestern University. e-mail: [email protected] Geometer’s Sketchpadr files for the diagrams in this paper are available from the author’s website http://www.math.fau.edu/Yiu/DynamicTriangleGeometry/DTG.htm. P. Yiu (&) Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, FL 33431-0991, USA e-mail: [email protected]
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Fig. 1 The centroid, circumcenter, and orthocenter
Fig. 2 The Gergonne and Nagel points
(4)
the incenter (I in Fig. 2) and the incircle with its points of tangency with the sidelines, and the Gergonne point (Ge in Fig. 2) which is the intersections of the three lines each joining a vertex to the point of tangency of the incircle with the opposite side,3
3
The triangle whose vertices are the points of tangency of the incircle with the sides is called the intouch triangle. See Fig. 2.
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Fig. 3 The lines La ðsÞ and La ðs aÞ
(5)
the excircles and their centers, with the points of tangency with the sidelines, and the associated Nagel point (Na in Fig. 2),
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