Entry tags:
Mediosimilar Triangles
I've been pretty vague about my research. Mostly this is because my actual goal is kind of meta. I'm not interested very much in individual classes of polygons, but rather in a theory of classes - what classes are, how to categorize them, and how different classes resemble or are related to each other. Describing that theory, that methodology, to non-mathematicians would be difficult.
What I can do, though, is discuss some of the fruits of that methodology - some of the classes of polygons that it has identified. So, this is the first of, probably, several posts on some isolated classes. I'll start off with what I call "mediosimiilar" triangles.
First, some notation and some things that are well-known to triangle geometers. If ABC is a triangle, we use "a" for the length of the side BC, "b" for AC, and "c" for AB. The "A-median" is the line (or line segment, depending on context) connecting A to the midpoint of BC; the B- and C-medians are defined similarly. The three medians intersect at a single point, the "centroid" of ABC, usually denoted "G". If you reflect the A-median in the bisector of the angle at A, you get the "A-symmedian", and likewise for the B- and C-symmedians. The three symmedians also intersect in a single point, called the "symmedian point", denoted "K".
If ABC is a triangle, there is a triangle DEF whose sides are parallel to and congruent to the medians of ABC: DE is parallel and congruent to the C-median, DF to the B-median, and EF to the A-median. I call DEF the "triangle of medians" of ABC. The triangle of medians of DEF is similar to ABC, with similarity factor 3/4.
So much is, as I said, well-known. Here's my contribution. ABC is "A-mediosimilar" if it is similar to DFE. Note the reversed order: AB is proportional to DF, AC to DE, BC to FE. "B-mediosimilar" and "C-mediosimilar" are defined similarly (using FED and EDF respectively). It turns out that mediosimilar triangles have a lot of interesting properties, which I discuss (among other things) in Taxonomy I. Then the following are equivalent: ABC is A-mediosimilar; 2 a^2 = b^2 + c^2; 2 d^2 = e^2 + f^2; the line through G and K (the "symmedian trail") is parallel to BC.
Being mediosimilar also interacts in interesting ways with other special lines of triangles, such as the Euler line (passing through G and the circumcenter of ABC), the Brocard axis (passing through K and the circumcenter), and the orthic axis (which I'm not going to try to describe - it's complicated). Mediosimilarity is a "sibling" of being isosceles, which yields a number of other interesting interactions. You can construct mediosimilar triangles easily: construct an equilateral triangle XBC, and draw the circle centered at the midpoint of BC and passing through X. If A is any point on that circle, ABC is A-mediosimilar (and it isn't if it's not).
So, this is the sort of thing that my methodology reveals. Is there any interest in more examples?
What I can do, though, is discuss some of the fruits of that methodology - some of the classes of polygons that it has identified. So, this is the first of, probably, several posts on some isolated classes. I'll start off with what I call "mediosimiilar" triangles.
First, some notation and some things that are well-known to triangle geometers. If ABC is a triangle, we use "a" for the length of the side BC, "b" for AC, and "c" for AB. The "A-median" is the line (or line segment, depending on context) connecting A to the midpoint of BC; the B- and C-medians are defined similarly. The three medians intersect at a single point, the "centroid" of ABC, usually denoted "G". If you reflect the A-median in the bisector of the angle at A, you get the "A-symmedian", and likewise for the B- and C-symmedians. The three symmedians also intersect in a single point, called the "symmedian point", denoted "K".
If ABC is a triangle, there is a triangle DEF whose sides are parallel to and congruent to the medians of ABC: DE is parallel and congruent to the C-median, DF to the B-median, and EF to the A-median. I call DEF the "triangle of medians" of ABC. The triangle of medians of DEF is similar to ABC, with similarity factor 3/4.
So much is, as I said, well-known. Here's my contribution. ABC is "A-mediosimilar" if it is similar to DFE. Note the reversed order: AB is proportional to DF, AC to DE, BC to FE. "B-mediosimilar" and "C-mediosimilar" are defined similarly (using FED and EDF respectively). It turns out that mediosimilar triangles have a lot of interesting properties, which I discuss (among other things) in Taxonomy I. Then the following are equivalent: ABC is A-mediosimilar; 2 a^2 = b^2 + c^2; 2 d^2 = e^2 + f^2; the line through G and K (the "symmedian trail") is parallel to BC.
Being mediosimilar also interacts in interesting ways with other special lines of triangles, such as the Euler line (passing through G and the circumcenter of ABC), the Brocard axis (passing through K and the circumcenter), and the orthic axis (which I'm not going to try to describe - it's complicated). Mediosimilarity is a "sibling" of being isosceles, which yields a number of other interesting interactions. You can construct mediosimilar triangles easily: construct an equilateral triangle XBC, and draw the circle centered at the midpoint of BC and passing through X. If A is any point on that circle, ABC is A-mediosimilar (and it isn't if it's not).
So, this is the sort of thing that my methodology reveals. Is there any interest in more examples?