@D.C.theIII Sure. I see what you did. It was ok, but introducing a new letter confused me.

Yea. I was just unsure if that new letter was going to have some other significance. But that is me overthinking things

I've been working on this question off and on usually waiting for your help, so with this part done. I was fiddling with how to express the area of a triangle with my angle measurements, but the ideas I came up with would include introducing new angles.

Writing it out.....actually no, I won't be introducing any new angle variables.

dropping the vertical down from an included angle would give me $\frac{\theta}{2}, \frac{\pi}{2}$ and $\frac{\pi}{2} - \theta$

ahhhhh victory.

So I can write "base" and "height" in terms of $\theta$ and $\gamma$

Wait. You’re fooling yourself again. If the triangle isn’t isosceles, the height doesn’t bisect the included angle. This is very standard. Don’t mess with the angle. Use one of your known sides as the base.

sigh...

@TedShifrin Hey Ted, you got 5 minutes to read a pretty small answer? I just wanna confirm if something is "intuitive" enough like I claim

@D.C.theIII I think what you're looking for is called Bharamagupta's formula for cyclic quadrilaterals

No, I went over that with Ted

hi

@SineoftheTime sup

@ペガサスSeiya how're you doing?

@SineoftheTime Doin great, just woke up. What about you?

@ペガサスSeiya I'm preparing an exam

Here it's 10:35 pm

@SineoftheTime same preparing for my next exam this morning. Japanese language....

@ペガサスSeiya I have topology next thursday

japanese must be tough

@SineoftheTime it is, kanji is really difficult

Hiragana too

Wish it was just a simple math exam instead

-_-

Can't wait to prove funny results after getting my degree

@ペガサスSeiya yeah, I prefer math exams even if are tough. My lowest mark was last year on a language exam

@SineoftheTime in the same boat. I never perform well in language exams, I can just barely manage to pass by studying all night. Mathematics is much easier to get a nice grade for

@ペガサスSeiya fortunately there's only one language exam in the first year

Wish I could change all my language exams with video games exams

@ペガサスSeiya lol

I'm gonna grab breakfast from my favorite restaurant today

@ペガサスSeiya Depends what it’s about, but sure.

@ペガサスSeiya video games won’t be helpful in real life.

@TedShifrin thanks, this is the one: math.stackexchange.com/a/4637246/1092912

@TedShifrin true, but they're fun after I've had a long day

Soooo ugly. You need to learn how to do better diagrams/pictures.

@TedShifrin yeah that's a different topic but I agree. Do you think the explanation is intuitive enough as I claimed?

i realize that details don't matter in figures like that, but, haha at that "equilateral triangle"

@leslietownes oh there's much worse than that weird equilateral triangle. I've seen equilateral triangles that look like obtuse triangles

known side as a base......height, how we characterize height?

@ペガサスSeiya No, I can’t follow it.

@D.C.theIII perpendicular distance from the opposite vertex to the (extended, if necessary) base

@TedShifrin which parts wasn't intuitive? I'll edit it and explain more. If you couldn't follow it chances are a large number of readers won't be able to either

First I had to work to see why the original triangle is isosceles. The cyclic quadrilateral stuff was a total leap. I know about them, but I don’t see it and your diagram is cluttered and hard to deal with.

@TedShifrin so you mean the first part where I talk about why $DK$ is equal to $BD$ and $DC$ using the properties of cyclic quadrilaterals. That's unclear right?

Yes. Write shorter, clear sentences. One idea at a time. Why cyclic? What inscribed angles?

@TedShifrin got it, I'm gonna draft up a new edit to make everything far clearer.

I don't know why I'm so fond of making such leaps without clarifying them first

Yiou weren't kidding when you said Math guys aren't great at writing. Shoulda taken those writing classes huh @TedShifrin

So I extended side $a$ and the green line is my distance from the opposite vertex. Is this the way you meant it?

That quadrilateral ain't looking very cyclic

No. Opposite vertex in the triangle! Extend the base the other way.

Poor Ted

nods

Being great at math aside, dealing with extremely inexperienced (for the lack of a better term) students daily must takes an entirely different set of skills

Teaching is an entirely different skill set.

Yeah I imagine being great at math alone isn't enough to be a good math teacher

That doesn't look right DC

growls Triangle with sides $a,b$ and included $\theta$.

Man this is painful

bruh....

Perspective for you, grasshopper.

chill with that noise. THis is why we're here to learn.

not you Ted, the dismissive comment before

@D.C.theIII I'm sorry if that was a little standoffish or rude. To be fair I'm not great at drawing either, just look at my conversation above with Ted

The grasshopper is Seiya.

There's a fine line between pleasure and pain during the learning process...

@user85795 no pain no gain

Pain starts to hurt after too much of it, pal.

@ペガサスSeiya I mean I get it, you know what to do, but before you knew how to do it there was a time when you had to be frustrated. Maybe not here, but elsewhere. So keep that in mind when things seem "easy" to you and may cause difficulty for another.

^

🤫

@D.C.theIII No I'm trying to say that even I don't know what to do sometimes, that's why I laugh at myself. Again, I'm sorry if you didn't like that.

There are absolutely times where I'm just staring at a problem and start to wonder what I'm doing with my life

entonation and emotion tend to get lost through text and misinterpretation can happen as a result. So do be mindful. It is all good, I know you didn't mean harm.

Seiya. You need to work on people skills. I do lose patience with some in here.

You're right. I act like this is Discord (the app), admittedly, and that gets me into trouble sometimes

As the old adage goes: if you have nothing good to say, then say nothing...

@TedShifrin grasshoppers can fly though

That said, we don’t all have to be infinitely patient here. But we should not be boldly rude.

Gotta go now, can't (shouldn't) text while driving

I had forgotten that “grasshopper” came from Kung Fu.

@ペガサスSeiya It's not about being a discord thing, it is about context. ANyways, it is long past. Everything is back to sunshines and lollipops..

when grasshopper can snatch the pebble from the master's hand he will be ready to leave the temple

i am having trouble understanding why rudin says $B_n$ is the union of a countable set of countable sets... My expectation is that $B_n$ is the union of an at most countable set of countable sets since I am seeing $B_n \sim \bigcup_i A_i$ - something like that

I mean $B_1$ is equivalent A so it is surely a finite union (namely no union?) of countable sets.

That is, I am identifying $B_n$ as the union of sets $A_i$ equivalent to $A$ (that is, countable) where the index $i$ denotes with what $b_i \in B_{n-1}$ the sets $A_i$ are associated with

e.g. $B_2 = A_1 \cup A_2$ where $A_1$ is the set of all 2-tuples $b_1, a \in A \sim A$, making $B_2$ a union of a finite (and therefore not countable) set of countable sets

oh wait... i think i see it

Each $B_{n-1}$ is a countable set, and for each $b\in B_{n-1}$, we have a countable set of $a\in A$. This is a countable union of countable sets.

Right, although I would leave the third side of the triangle in the picture. So what is $1/2$ base times height?

So I'm not going to be all high and mighty, but I knew this is what you were intimating at, I didn't just jump to it immediately and what is still troubling me now is that I can't see how I would write $a$ in terms of $\theta$. I see how how the height will be written in terms of $\theta$, but not my base

This formula has shown up in various exercises before in the book, by the way. There's one for sure in the early sections of Chapter 3.

The base is $a$. What's the height?

the third side of the triangle meaning the one that "splits" my original quadrilateral? Height will be: $b \sin(\theta - \frac{\pi}{2})$

Right. So we can see the triangle whose area we are computing :)

The formula (which you learned in trig and probably haven't used enough) is $\frac12 ab\sin\theta$.

oh shoot your right. $\theta - \frac{\pi}{2}$ is the angle for the third side

@TedShifrin Lol...if ever.....I recognize it, but I don't think I ever used it consiously as the area formula

This very formula was needed to do #11 in the same section. If I didn't assign that, I probably did it in lecture.

yea not assigned

It comes in also with the area of the parallelogram that appears in the magnitude of the cross-product, although I purposely did not do that in the book.

But if you ask most any calc III or physics student, they'll tell you $\|\vec a\times\vec b\| = \|\vec a\| \|\vec b\|\sin\theta$.

It clicked why I don't need to express $a$ in terms of $\theta$, it is already a fixed constant. It was only height that needed recharacterizing.

Yuppers. Of course, if you decide to use $b$ as the base, then the height is $a\sin\theta$. No problem.

I enjoy your geometry questions even though this probably should not have taken as long as it did. My faulty (albeit gradually improving) geometry is what is hindering me

No argument from I = me.

It's irksome, but at least I am at ease having a picture in mind instead of just symbol pushing with no meaning......

@TedShifrin 😭

ok, time to write out an "outline" of the procedure to do it tomorrow, then a late lunch, then finish off a question establishing QR decomposition in linear algebra

QR is good numerically, too. There's an amazing algorithm based on it to numerically find the eigenvalues/eigenvectors of a matrix. You do $QR$, switch to $RQ$ and repeat and repeat ....

@TedShifrin Your book(s) aren't about synthetic geometry (where I think cyclic quadrilaterals belong) so, how did this problem even come up?

Cyclic quadrilaterals show up in complex analysis and projective geometry a little bit. This was a famous constrained max/min problem.

Yeah I'm interested in it because I had seen it years ago when I did a linear algebra course, but it wasn't touched upon. Then in my linear regression course one of the graduate texts which was recommended as an accompanying text had the derivation of the least squares estimators. They had one derivation using multivariable calc and then they had the QR Decompostion and mentioned that in practice the QR decomposition is what is used in software.

By the way, you should learn some perspective on geometry. Get Pedoe's wonderful book, Geometry: A Comprehensive Course. Published by Dover and cheap. @Seiya

I always touch on QR because it's the Gram-Schmidt algorithm, which is actually important (as a theoretical tool) in geometry.

I have been meaning to get an accompanying geometry text. Even if I don't read it all the time, something to just constantly read to cement things over time

That by the way was to Seiya, not you.

oh....so stick to my high school review texts for now....🙃🙃

I'm definitely not recommending that book to you.

Yeah, you're not heading in particularly geometric directions in your need for multivariable calculus/linear algebra.

@TedShifrin Can I get hard copies delivered for it? What's it about it? Must be interesting if you're recommending a book

until I am the one that writes the world changing thesis that brings the two subjects together.