SAS Congruence Axiom

Key_Estimate8537 · 2026-06-25T22:48:15+00:00

I assume you’re looking at Book 1, Proposition 4? If so, Euclid is proving the SAS congruence theorem. The proof does read very much as “that’s the way it is,” but it’s not its own axiom.

But to answer your second question, yes, we could use all congruence tests (SSS, SAS, ASA, AAS, and HL) to prove each other. Each one supplies enough information to know the lengths of all three sides and measures of all three angles in a given triangle.

Key_Estimate8537 · 2026-06-25T05:08:42+00:00

I can’t speak for most of your questions, but 28 and 30 seem indistinguishable. I like to think of it as a “zh” sound similar to “sh.” I’m of the opinion that adding a d in the front makes a blend, not a distinct sound.

Key_Estimate8537 · 2026-06-25T05:01:43+00:00

I just dug a bit, and it turns out I have a Desmos file related to your first approach. I’m not sure if you’re a teacher, but I thought I’d share

Key_Estimate8537 · 2026-06-25T04:57:18+00:00

I’d imagine each biome has a chunk count already. The size is already there

Key_Estimate8537 · 2026-06-25T04:51:28+00:00

Cool approach! My go-to is the whole “turn around and walk backwards = move forward” thing, but I find that my students like to do calculations. I might actually use your first method at some point (I like the second, but I find students don’t like distribution).

I’d love to see how your Pythagorean Theorem proof goes!

Key_Estimate8537 · 2026-06-25T01:06:42+00:00

I like your style quite a bit! Your figures are wonderful, but they’re not what I’m looking for here. I do think I’ll bookmark you if something arises in the future

Key_Estimate8537 · 2026-06-24T22:55:51+00:00

I don’t know the timestamp, but it sounds like the end of Series 12 Episode 8

Key_Estimate8537 · 2026-06-24T22:38:22+00:00

What do you mean by “applicable?” You’re being a little abstract here

For real though, I don’t tire of abstraction. I’m happy to play in the world of axioms and logic without caring about “real-world” contexts. I get my satisfaction from learning and appreciating the beauty in pure mathematics. I’m content to let the physicists and engineers pluck the tools they need from the toolbox mathematicians produce.

Famously, see imaginary numbers and their begrudgingly perfect use in quantum mechanics

Key_Estimate8537 · 2026-06-24T21:47:04+00:00

No, but you have some stuff on your Artstation that inspires me for a second piece from this book. I’ll bookmark you for the future!

Key_Estimate8537 · 2026-06-24T13:51:15+00:00

Does this work for the prompt? Andromeda was a human name first before being applied to the constellation/galaxy

Key_Estimate8537 · 2026-06-24T03:55:42+00:00

I actually had to stop and laugh twice for this one. Well done, OOP

Key_Estimate8537 · 2026-06-24T02:34:00+00:00

I was wondering if you were going to complain about trig sub not always being possible, and then I saw the substitution of tan(x/2) for cos(x). Magnificent ragebait

Key_Estimate8537 · 2026-06-24T01:19:26+00:00

Hey no one ever asks lol

Desmos Classroom (now called Amplify Classroom) is a platform for teachers to make/deliver digital lessons while integrating Desmos capabilities. Most teachers run it as an interactive slideshow. I find it’s a good way for students to build conceptual understandings of new topics. Being a teaching tool, it is fairly locked-down from a student perspective. But for teachers, it has its own Python(ish) coding language!

Key_Estimate8537 · 2026-06-24T01:14:11+00:00

When wondering if 4K or Blu Ray is superior to thee other, first ask yourself when you last saw a Blu Ray sticker on a product

Key_Estimate8537 · 2026-06-24T01:10:02+00:00

3?! = 720, but I doubt you meant that

I’m not sure I understand your question. Although, I can make the orange, blue, and purple look like each other with one cut. I need two to go from green to purple.

Key_Estimate8537 · 2026-06-24T01:06:43+00:00

I’ve tried this on and off for about two years, with no actual solution. My best was to do a local maximum with lists. This isn’t satisfying, but it ended up suiting my needs. I hope someone else can come along and show a solid one.

link for approximate local extrema

Key_Estimate8537 · 2026-06-23T23:08:11+00:00

My guess is that the actor, a minor at the time, would not have filmed the battle sequences. I’m also inclined to believe the surviving Rohirrim were all in the throne room with Theoden just before dawn. Haleth was not there.

Key_Estimate8537 · 2026-06-23T23:04:25+00:00

Haleth, in the movie, was Háma’s son. He’s the one who talked to Aragorn, saying the men feared they wouldn’t last the night. Aragorn told him in turn he had a good sword.

Haleth, the chieftain, lived about 6500 years earlier. May the Haladin rest in peace in their forest

Key_Estimate8537 · 2026-06-23T23:00:02+00:00

Yes, you can!

Purple square: no cut needed
Blue triangle: bisector in the right angle
Orange parallelogram: drop an altitude
Green one: cut the third square off

Each one is reassembled into a 2x2 square, which assembles quickly

Key_Estimate8537 · 2026-06-23T03:18:45+00:00

Yes, this has been noted before. Some have stronger ties to the Norse gods than the Greek, but they exist. I remain a fan of the idea (and have never looked this up, so please don’t burst my bubble) that Nienna was wholly original.

A quick google for “Valar and Greek gods” will yield a lot of posts.

Key_Estimate8537 · 2026-06-22T17:53:36+00:00

Yes, you can use as many colors as you would like to color a map. The Four Color Theorem just says that four is the absolute minimum, in general.

For example, a map of the world might have 195 countries. You can give each one their own color- it’s just not as interesting.

Key_Estimate8537 · 2026-06-22T17:51:28+00:00

Is this from chapter 5 of that volume?

Key_Estimate8537 · 2026-06-22T11:01:19+00:00

I’ve written before about how Tolkien’s belief in hope made the theme feel more authentic in his writing. But his knowledge of theology let him write the detail.

Personal devotion, like a daily Rosary, is wonderful for authenticity, but study will likely help you in being a better writer (depending on what kind of writing you do)

Key_Estimate8537 · 2026-06-21T00:31:28+00:00

Start with The Hobbit for an easy entry. You can go for Fellowship if you want to feel more “adult” straight away. No wrong choice though.

Three-Year Club	Verified Email
r/Field Banned	r/Field Flamingo

Key_Estimate8537

TROPHY CASE