Mediosimilar Triangles
Feb. 8th, 2019 08:07 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowI'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?
