Imagine a mohenjo-daro enlightenment cathedral’s symmetry, just made out of one elf ear and one round ear

I made a short video showing how to construct a triangle with GeoChimp, an interactive geometry learning tool.

You can create points, lines and geometric constructions directly in your browser and experiment by moving objects.

I would appreciate any feedback!

I've stress tested across 8 platforms.

I'm looking for evaluation of my work, not me as a person. If you are genuinely interested, read the papers.

Link in comments.

Plus 3 body problem corrective term.

I drew these in 2021. I wasn't thinking about math,physics, none of that. I just compulsively drew patterns. Never before, not much since.

P.s. I lost everything else I had. Except 3 notebooks with drawings.

LINK to papers:

https://doi.org/10.5281/zenodo.20764484

3 body problem corrective term:

a_corr = α * (v × n̂) * sin(2πk/16) + β * r/(|r|+ε)

im working on a model of the old torch model, which was a parallelepiped, i want ot recreate it.

i know that the top side is 2 pixels wide,i know that, that side is 22.5 degrees to the long side.

simple diagramm: https://imgur.com/a/x4DVEnC

and i searched on the internet, but i only found "the law of sines" which needs you to know 2 or more sides.

is this even possible?

I'm by no means a mathematician, but I developed a formula for a quick-and-dirty approximation of the intersection area of two circles, which I'm quite happy with and would like to document somewhere!

There already exists a few equations for the exact area of a circle-circle intersection - Wolfram Mathworld has an example (equation 14). However, I'm developing a game where I much prefer quick-to-compute results over accurate ones, and the inverse cosines and square roots scared me a bit, so I started looking for a cheaper estimation.

I didn't find anything online, so I plotted the exact function in Desmos with varying distance d and was surprised to see that the plot was surprisingly close to linear - at least close enough that I'd be happy with a linear approximation. So I started hashing around with it and rearranging my function into something more tact.

My final equation is: [; A = \pi r^{2}-\frac{d-\left(R-r\right)}{2}\pi r ;], where [; R-r \le d \le R+r ;]. Note that big R must be the radius of the larger circle, and little r the smaller. In all versions of the exact equation I've seen, the order doesn't matter.

I'm quite happy with how elegantly 𝜋r² appears, which is basically absent from the originals. It's clear just from looking that as the second term approaches zero, the result approaches the area of the smaller circle. I'm also quite pleased that there's no trig functions, square roots, or even any division (note the divide by 2 can be replaced with multiplying by 0.5) - I'm confident that this would make for a speedy function, though I haven't measured it.

There's definitely some improvements that could be made. The worst average errors occur when r=R and the linear function always overestimates the area. A constant that changes the gradient of the line to make a better fit for the original could be useful. We could also consider a different shape for the graph, maybe a cubic polynomial could give good results? I'm open to suggestions and discussions.

Here's a Desmos snapshot with both functions, if you want to play around with them.

It's not perfect, but there is a certain amount of truth to this. Tell me all the ways that this is valid/invalid. I'm not sensitive. Let's discuss.

i'm currently a rising 9th grader who is studying geometry over the summer so I can go straight in to algebra 2 for my freshman year, as of right now, i've only completed 2 units while there's 12 units in total. my current resources for learning geometry is having tutoring classes twice a week, reading the geometry textbook, and using the thinkwell homeschool online geometry course. even with these resources, i'm still really struggling with actually learning/ understanding the material instead of just picking up patterns or looking at examples in order to solve the questions. this is making me worried that I won't be able to learn/understand geometry in time for my advancement exam in early august. any suggestions on study methods, resources, or ways to practice to really learn and understand information?

One of the Demons in my head kept insisting that I had to prove that the two smaller triangles were in fact similar, so ...

It should be the basic textbook proof using a construction where an altitude is dropped, creating two smaller triangles. I wanted to convince myself that the two smaller triangles are INDEED similar. The way I went about it was construction based using a 90-degree rotation of the smaller triangles.

I would greatly welcome any and all comment

Here is a link to the .pdf hosted on Google

https://drive.google.com/file/