For a given triangle, we consider several sequences of nested triangles
obtained via iterative procedures. We are interested in the limiting behavior
of these sequences. We briefly mention the relevant known results and prove
that the triangle determined by the feet of the angle bisectors converges in
shape towards an equilateral one. This solves a problem raised by Trimble~[5].

On sequences of nested triangles

L. Richard Hitt received his B.S., M.S., and Ph.D. from Florida State University. He is a professor of mathematics at the University of South Alabama where he has been on the faculty since 1977. His research interests include geometric topology, especially knot theory, as well as elementary and differential geometry. His teaching interests also include mathematical modeling and the use of computing technology in mathematics.

https://www.ems-ph.org/journals/show_pdf.php?issn=0013-6018&vol=56&iss=1&rank=4

Dynamic Geometry of Polygons

“Mathematics is about problems, and problems must be made the focus of a students mathematical life. Painful and creatively frustrating as it may be, students and their teachers should at all times be engaged in the process - having ideas, not having ideas, discovering patterns, making conjectures, constructing examples and counterexamples, devising arguments, and critiquing each other’s work.“ ‐ Lockhart (2002, p. 16) Lockhart (2002, p. 2), a research mathematician, describes the present system of mathematics education at school as a ‘nightmare’ claiming that it destroys children’s ‘natural curiosity and love of pattern-making’. He goes further in his critique of school mathematics by saying that ‘no actual mathematics’ (p. 14) is being done at school, and in the ‘place of discovery and exploration’ (p. 15) there is only mindless drill and exercise of given rules and algorithms.

/pdf/conjecturing-refuting-and-proving-within-the-context-of-26x3zx51k1.pdf

Conjecturing, refuting and proving within the context of dynamic geometry

We introduce a concept "degree of triangle centers", and give a formula expressing the degree of triangle centers on generalized Euler lines. This generalizes the well known 2 : 1 point configuration on the Euler line. We also introduce a natural family of triangle centers based on the Ceva conjugate and the isotomic conjugate. This family contains many famous triangle centers, and we conjecture that the de- gree of triangle centers in this family always takes the form ( 2) k for some k 2 Z. MSC 2000: 51M05 (primary), 51A20 (secondary)

/pdf/degree-of-triangle-centers-and-a-generalization-of-the-euler-3blpn1jv01.pdf

Degree of Triangle Centers and a Generalization of the Euler Line

When considering ‘convergence’ many people think of number sequences or contexts arising from calculus. But there are also interesting phenomena of convergence – easy to visualize – arising in elementary geometry. Some of them are so elementary that they can be dealt with at school, for instance an example of iteration that is described in [1, p. 171f] (see also [2, p. 59], [3, p. 222f] and [4, p. 42ff]).

S. Y. Trimble

Papers

80.42 The limiting case of triangles formed by angle bisectors