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uhhh does anyone know how to solve angle & perpendicular bisectors?
i kinda got the gist of it (as i went through notes and watched videos) but i need it to be re-explained to me as if im a toddler
if you’re that person who saw this and went: ‘oh this shit’s easy’ then can you do me a favor and tell me how to solve them in simple words please,,math isn’t my expertise
Irregular polygons are awesome. You can do so much more with them, in so many areas, but I'm going to limit myself to talking about just two of them: tiling and triangle centers.
(Did you think I was going to be all snobby toward my irregular friends and give them only grudging approval? Heck no, I love those guys! And so by the rules of the meme I get to infodump about the things I love, so this may be long but you asked for it.)
Let's first talk about covering the plane with copies of a single shape -- a monohedral tiling. And for now, let's restrict ourselves to periodic tilings. All triangles and all non-self-intersecting quadrilaterals tile the plane periodically, so that's not very interesting. All you have to do is place one polygon and then make copies by rotating 180 degrees around the midpoints of the sides.
With five sides, things become more complicated, because regular pentagons don't tile by themselves, but there are fifteen ways an irregular pentagon can periodically tile the plane. Here are four of them that were discovered in 1976 and 1977 by Marjorie Rice, an amateur mathematician.
There are three types of monohedral periodic convex hexagonal tilings.
For polygons with seven or more sides, there are no monohedral periodic tilings using a convex prototile , but there are periodic tilings for nonconvex polygons of any size. Some of them are quite famous.
Most of Escher's work in his Regular Division of the Plane series uses shapes with curves as well as straight sides, so they don't show polygon tilings, strictly, but the patterns do point toward complex tilings that are visually pleasing.
Irregulars can tile aperiodically, too. Here's a pentagon tiling with 6-fold rotational symmetry. It can be extended infinitely, and tilings can be constructed with pentagons for n-fold symmetry of any n>2.
Going back to non-convex shapes, here's the Voderberg tile, an enneagon that forms a spiral tiling. Notably, one copy of the shape can be completely surrounded by two others.
And of course I can't go without mentioning the tiling news of the century: "Tile (1,1)", aka the Hat, aka the T-shirt, a tridecagon (and polykite) that can tile the plane but only aperiodically. IDK if you follow polygon news but this was huge.
Okay, enough about tilings. With tilngs it's pretty easy to get what's going on just by looking at them, but my next topic, triangle centers, requires a bit more explanation. Also there's a bit of jargon, but I will try to keep it simple.
Take an arbitrary triangle ABC. Where is its center? One way you might define it is to find the midpoint of each side and draw a line to it from the opposite vertex. Each line divides the triangle in half, and these three lines (the medians) all cross at a point, the centroid. This works for any triangle, no matter its shape. The point marks the center of gravity of the area of the triangle, and also the center of gravity of its vertices. Based on that, you could consider this the center of the triangle.
Or you could work with angles instead of sides. Draw lines from each vertex that divide the angles in half (the angle bisectors). These all meet at a point called the incenter, which marks the center of the largest circle that fits inside the triangle. To put it another way, it's the point that is equidistant from all three sides. That's another point you could call the center of the triangle.
Or, how about a circle around the triangle instead? From the midpoints of the sides, draw the perpendicular bisectors. Again, they all intersect at a point, the circumcenter, which is the center of the circle that passes through the vertices -- the point that is equidistant from all three of them. So you could also call that the center of the triangle.
Or how about drawing perpendicular lines from the sides again, but having them pass through the opposite vertices (the altitudes)? They coincide at a point called the orthocenter. Isn't that neat? Yet another point we could call the center of the triangle.
But wait -- can we? For an obtuse triangle, the circumcenter and the orthocenter are going to lie outside the triangle. (For the orthocenter of an obtuse triangle, you have to extend each side into a line, and draw the altitude as a perpendicular to that.) Being outside a thing is really not what we have in mind when we talk about the center of the thing. Should we care about that?
Maybe not. Check this out. We'll go back to the circumcircle, and draw tangents to it at the three vertices. The three tangents form the tangential triangle (in blue), which we'll call A'B'C', where A' is opposite A, and so on with B and C. Now draw the circle that passes through A, A', and the circumcenter, and do the analogous construction for B and C (in red). The three circles coincide in two places: the circumcenter and another point called the far-out point. And as the name suggests, this is usually well outside the triangle, even for acute triangles.
There's no reasonable way to call this point the center of the circle. But so what? I just love the fact that the three circles line up like that. I no longer care about finding "the" center of the triangle. I no longer care that "center" is rather a misnomer for many of these points. I just think it's neat that you can draw these constructions on ordinary aysmmetrical triangles and they keep all converging on one point. Want more? Reflect the medians across the angle bisectors, and they all meet at the symmedian point. Or connect the vertices of the tangential triangle with the intersections of the medians and the circumcircle. Those lines meet at the Exeter point. Or, from each vertex, draw the line that splits the perimeter of the triangle in half. These are called the splitters, and they meet at the Nagel point. And on and on and on.
You can simply wander around a triangle, connecting things that relate somehow to vertex A, then do the equivalent thing for B and C, and stumble upon new centers. And there are tens of thousands of these things, constructed with straightedge and compass or by other methods. And there are so many ways to enjoy these things. You can page through the enormous collection and get a kind of stamp-collecting satisfaction just looking at their variety and knowing that they exist. Or you can appreciate the proofs that show that the constructions really do specify a unique point. Or proofs that show that a point constructed to have one property has a surprisingly different property. Or you can notice that the points fall into certain families and appreciate the connections between them. (For instance, the centroid, circumcenter, orthocenter, far-out point, and Exeter point, among others, all happen to lie on the same line, the Euler line.) Or you can convert the points to trilinear coordinates, manipulate them algebraically, and get to know them that way.
But to appreciate them at all, you need to work with irregular triangles. Because here's the thing: in an equilateral triangle, all these points collapse to the same point. Everything simplifies to a single center, and the incredible wealth of invisible structure that teems inside every ordinary triangle is gone.
You have finally found "the" center, but at what cost? Symmetry is death. Only through asymmetry will the vast truth of the triangle be revealed to you.
And those are just a few of the reasons irregular polygons are cool!
hi, I had a question. On your Jax art there are Spanish lyrics(?) In the background. I was wondering what song it was? Or where it's from? Sorry I'm just curious. Absolutely love the art
Thank you ! I’m so happy you like the art and that you’re curious about the song - it’s called “Así Soy Yo “ (That’s How I am )
by El Cuarteto de Nos
Here’s the translated lyrics if you’re interested:
My mom got me a bunny! I named her Sherbet because she sort of reminds me of sherbet. I went through several names however. At first it was Bun-Bun and then Hop-Hop and then BeBe and then Poppy, and then I think I finally settled on Sherbet. I will let you know if I change my mind again. Isn't she cute though?
The study day went by quickly. I didn't need my full four hours, so I added in a supplement for Biology since we're on the evolution unit right now. My curriculum is mostly a creationist perspective, although it touches on evolution and natural selection, so I found this docuseries called Life on Our Planet which dives deeper into evolution. I don't need my parents permission to watch the docuseries since they give me a lot of freedom around things like that, but I still let them know I was going to. Apparently my dad has already seen it, and he thought I would enjoy it. That was a surprise to me that he had watched it. The first episode was suspenseful, but I am looking forward to watching more. I will have to build it into my study day somehow. I think it has 10 episodes, all a little under an hour long.
Tasks Completed:
Geometry - Learned to construct perpendicular bisectors and angle bisectors + practice + practice with proofs involving segments and angles + honors work
Lit and Comp II - Reviewed Unit 21 vocabulary + read chapter 39 of Emma by Jane Austen
Spanish 2 - Vocabulary test (100%)
Bible I - Read Judges 12-13
World History - Read second part of the introduction to the unit + read about Albert Einstein + read about Pablo Picasso + viewed some of Picasso's artwork
Biology with Lab - Read and watched videos on the functionality of life from a creationist perspective
Foundations - Read the definition of self-control + completed next quiz on Read Theory + read an article about propaganda in schools
Piano - 60-minute piano lesson + practiced for one hour
Khan Academy - None today
CLEP - None today
Streaming - Watched episode 1 of Life on Our Planet (evolutionist perspective)
Reading - Read pages 177-226 of House of Salt and Sorrows by Erin A. Craig
Chores - Laundry
Activities of the Day:
Personal Bible Study (Colossians 3)
Ballet
Pointe
Journal/Mindfulness
-
What I’m Grateful for Today:
I am grateful that my parents agree and encourage me to study the beginnings of life from creationist and evolutionist perspectives instead of only creationism.
Quote of the Day:
Every act of creation is first of all an act of destruction.
-Pablo Picasso
🎧Suite in B minor for piano, op. 58 - Charles-Marie Widor
Видео взято с канала INKLING на YouTube. Видео называется "Мемы по Отелю Хазбин / 4 часть". Решила поделиться, в первый раз я так смеялась😅 Мне понравилось
The video is taken from INKLING's YouTube channel. The video is called "Мемы по Отелю Хазбин / 4 часть". I decided to share it, the first time I laughed so much😅 I enjoyed it
Translation, if someone needs it
Velvette: Am I a baby?!
Vox: Yeah. I'm 800-some dick years old
Valentino: Dicks are good) Do you have a big one?
Vox: Yes))
Valentino: How much?
Vox: 4 by 4
Velvette: Is it a square?
VVV: *laughing*
Vox: The fucking rectangle. AHHAHA
Valentino: You need to suck dicks, not find perimeters in them
VVV: *laughing*
Vox: What's yours?
Velvette: The fucking triangular
Vox: HAHA
Valentino: Is it also necessary to calculate the bisector before sucking?
Had to remind myself what that means, the reflection of the median across the angle bisectors (isogonal to the medians). And just like the medians are concurrent, then so are the symmedians, and their intersection is the isogonal conjugate to the centroid (the intersection of the medians).
There's an idea called the trilinear coordinates for a point in a triangle, and the trilinear coordinates for the symmedian intersection are the inverse of that for the centroid. I'm not sure how to produce the trilinear coordinates given a point, nor find a point from the trilinear coordinates. There's always more to learn.
This sketch generalizes the median to the corresponding points parameterized by t, and those segments isogonal segments. Pretty fascinating to me.
I don't know how to do my science work(I thought I'd ask for my father's help) but I got glared at for listening to music and drawing and not doing my work(that I don't know how to do) so I'm sitting here, tears in my eyes, constructing angle bisectors because its oddly charming as an activity for boredom while a girl watches cocomelon or something next to me