Anyone understand inversions? | Mathematics | Page 1

quartz folioBOT Oct 10, 2022, 12:49 AM

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worldly kayak Oct 10, 2022, 12:59 AM

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do you understand what a means

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which parts do you not understand

ruby juniper Oct 10, 2022, 1:01 AM

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worldly kayak do you understand what a means

Honestly all of it. I think "part a" says that If we invert a point on a line containing C, the inverted point will end up on that line

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That's the only part I have somewhat of an idea

worldly kayak Oct 10, 2022, 1:03 AM

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ruby juniper Honestly all of it. I think "part a" says that If we invert a point on a line co...

yep

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so basically if you have a line containing C

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and you invert all of the points on the line

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you just get the line again

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does that make sense?

ruby juniper Oct 10, 2022, 1:05 AM

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worldly kayak does that make sense?

I thought it's if you invert a point on the line, it just ends up somewhere else on the line

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Like this

Screen_Shot_2022-10-09_at_7.28.58_PM.png

worldly kayak Oct 10, 2022, 1:05 AM

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yeah that's correct

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what this theorem is saying is that the line itself remains unchanged

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so if you take every point on the line and invert it

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you get the line again

ruby juniper Oct 10, 2022, 1:06 AM

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Does it have to end up at a specific part of the line or can it end up anywhere?

worldly kayak Oct 10, 2022, 1:06 AM

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the theorem doesn't specify, so it can end up anywhere

ruby juniper Oct 10, 2022, 1:07 AM

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So is an inversion just taking a point in the circle outside and then vice versa (outside to inside?)

worldly kayak Oct 10, 2022, 1:07 AM

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yea

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in a specific way

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so that CP * CP' = r^2

ruby juniper Oct 10, 2022, 1:08 AM

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worldly kayak so that CP * CP' = r^2

Gotcha. Ok so what about part B? straight lines not containing C onto circles through C

Like is point P not on the line?

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Or you could explain it in your own words

worldly kayak Oct 10, 2022, 1:09 AM

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well suppose you have a line

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and the line doesn't go through C

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and then you take every single point on that line and apply the inversion to it

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then the resulting image you get is a circle that goes through C

ruby juniper Oct 10, 2022, 1:13 AM

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worldly kayak then the resulting image you get is a circle that goes through C

This isn't a perfect image, but something like this? Like the inversion of a point P will be a point on the circle when we invert all the points on that line?

Also how could we invert the points on the line (in the circle) when I thought inverstions were taking a point inside the circle to out and vice versa. We can invert a point inside the circle to inside it again

Screen_Shot_2022-10-09_at_8.10.42_PM.png

worldly kayak Oct 10, 2022, 1:14 AM

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ruby juniper This isn't a perfect image, but something like this? Like the inversion of a poi...

yeah you are right, points inside the circle invert to points outside the circle, so the circle you draw here is not quite right, but this is the right idea

ruby juniper Oct 10, 2022, 1:15 AM

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worldly kayak yeah you are right, points inside the circle invert to points outside the circle...

Would the line have to be outside the circle?

worldly kayak Oct 10, 2022, 1:16 AM

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yeah, if it inverted to that circle

ruby juniper Oct 10, 2022, 1:17 AM

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worldly kayak yeah, if it inverted to that circle

Ok just a quick question, I thought there need to be a ray between the center and the point we plan on inverting?

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Like this

Screen_Shot_2022-10-09_at_8.17.49_PM.png

worldly kayak Oct 10, 2022, 1:18 AM

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the idea is for each point on the original line, you draw the ray and figure out where that point goes

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and you do this for each point on the line

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(obviously there's an infinite number of points on the line, so you can't physically do this, but that's what the process of inverting the line is)

ruby juniper Oct 10, 2022, 1:22 AM

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worldly kayak the idea is for each point on the original line, you draw the ray and figure out...

So something like this? I just drew P and Q and picked random parts of the circle for them to go through

Screen_Shot_2022-10-09_at_8.21.17_PM.png

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So essentially an inversion is taking a point inside the circle, drawing a ray to it somewhere outside the circle (and vice versa)

worldly kayak Oct 10, 2022, 1:24 AM

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here this video might help you visualize it https://youtu.be/Z6GG8zsMWH8?t=119

YouTube

Mathemaniac

Problem of Apollonius - what does it teach us about problem solving?

This video uses the problem of Apollonius as a way to introduce circle inversion and an important problem-solving technique - transforming a hard problem into a simpler one; then solve for the simpler, transformed version of the problem before doing the inverse transformation so that we obtain the solution to the hard problem. This problem-solvi...

▶ Play video

worldly kayak Oct 10, 2022, 1:24 AM

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ruby juniper So essentially an inversion is taking a point inside the circle, drawing a ray t...

yeah exactly

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and the ray goes through the center of the circle

ruby juniper Oct 10, 2022, 1:28 AM

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worldly kayak and the ray goes through the center of the circle

Gotcha. So now we're at part C. Circles through C onto straight lines not containing C

This is the vice versa of part b 🤔

So if we invert any point on the circle that goes through C, we get a straight line not containing C (so the straight line is outside of the circle)

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I think that's what it is saying

worldly kayak Oct 10, 2022, 1:29 AM

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ruby juniper Gotcha. So now we're at part C. Circles through C onto straight lines not contai...

the straight line not containing C doesn't necessarily mean it's outside the circle

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but yeah if you have a circle through C, then it inverts to a straight line that does not contain C

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and yup it's the opposite as b

ruby juniper Oct 10, 2022, 1:30 AM

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But if we invert a point on any circle through C, then we need to be outside of the circle. Isn't that was inversions do?

worldly kayak Oct 10, 2022, 1:31 AM

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if the circle is through C, then not all the points on the circle have to be inside the original circle

ruby juniper Oct 10, 2022, 1:32 AM

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Like this is what I'm thinking

Screen_Shot_2022-10-09_at_8.31.47_PM.png

worldly kayak Oct 10, 2022, 1:32 AM

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if the circle that you drew is entirely contained within your original circle, then yea your line will be entirely outside of the original circle

worldly kayak Oct 10, 2022, 1:32 AM

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ruby juniper Like this is what I'm thinking

when you draw inversions, make sure your ray starts at C

ruby juniper Oct 10, 2022, 1:33 AM

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Good catch

Screen_Shot_2022-10-09_at_8.32.53_PM.png

worldly kayak Oct 10, 2022, 1:33 AM

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but yeah that's right, if your blue circle is like that then the resulting line will be outside of the purple circle

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but if you make your blue circle bigger so that parts of it are outside the purple circle, then your red line will be partially inside the purple circle

ruby juniper Oct 10, 2022, 1:34 AM

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worldly kayak but if you make your blue circle bigger so that parts of it are outside the purp...

But the points will be outside the purple circle (like always?)

worldly kayak Oct 10, 2022, 1:34 AM

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if hypothetically, the point P on your blue circle is outside the purple circle, then P' will be inside the purple circle

ruby juniper Oct 10, 2022, 1:43 AM

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Ok let me gather my thoughts, one sec, this is a lot to digest

ruby juniper Oct 10, 2022, 1:46 AM

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worldly kayak if hypothetically, the point P on your blue circle is outside the purple circle,...

So if I invert a point P on the blue circle that goes through the center of the purple circle, it can end up in the purple circle

and eventually every point on the blue circle that's inverted will create a straight line not containing C

isn't it possible for P' to be outside the purple circle? it just has to be outside of the blue circle. Just making sure, since inversions are transforming points outside circles and inside circles

Screen_Shot_2022-10-09_at_8.44.28_PM.png

worldly kayak Oct 10, 2022, 1:48 AM

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ruby juniper So if I invert a point P on the blue circle that goes through the center of the ...

if I invert a point P on the blue circle, it can end up in the purple circle
yup, this is correct! although P' is not in the correct place in the picture you drew (it should be between C and P)
every point on the blue circle will invert to become a straight line not containing C
yup!
isn't it possible for P' to be outside the purple circle?
yup, if P is inside the purple circle

ruby juniper Oct 10, 2022, 1:49 AM

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And then circles not through C onto circles not through C?

So like inverting points on a circle that don't go through C will just create another circle that doesn't go through C

worldly kayak Oct 10, 2022, 1:49 AM

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yup

ruby juniper Oct 10, 2022, 1:52 AM

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worldly kayak yup

And then this moreover part... So if you invert points from a circle, which create a straight line (or vice-versa), then the line joining C to the center of p... and this is where I get confused... if C is our center, where does the center "p" come from?

Screen_Shot_2022-10-09_at_8.51.39_PM.png

worldly kayak Oct 10, 2022, 1:53 AM

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there are two different circles here

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there is a circle k (with center C) and a circle p (with its own center)

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not sure what m is though?

ruby juniper Oct 10, 2022, 1:58 AM

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worldly kayak not sure what m is though?

When the inversion transforms a circle p into a straight line m, or vice versa, then the straight line that joins C to the center ofthe circle p is perpendicular to the given straight line m. When the inversion transforms a circle into a circle, their centers are collinear with C.

The book my prof uses (which I found online, we have no book for the class) explains it a little differently.^^

I'm confused how there are "two circles here". Ick has a circle with a center C... Isn't that what circle P is? we're inverting points on circle P with the center C

worldly kayak Oct 10, 2022, 2:00 AM

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ruby juniper So if I invert a point P on the blue circle that goes through the center of the ...

like for example here the purple circle would be your circle k (with center C) and your blue circle would be a circle p (with its own center)

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and then the blue circle transforms into a line m

ruby juniper Oct 10, 2022, 2:01 AM

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I thought k is the radius?

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Ick is where c is the center and k is the radius

worldly kayak Oct 10, 2022, 2:04 AM

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oh k is the radius?

ruby juniper Oct 10, 2022, 2:04 AM

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Yea

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I'm pretty sure

worldly kayak Oct 10, 2022, 2:04 AM

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then the purple circle would be your circle with center C and radius k, and your blue circle would be a circle p (with its own center)

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and then the blue circle transforms into a line m

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sorry, I wasn't sure what your prof's notation meant haha, should've asked

ruby juniper Oct 10, 2022, 2:05 AM

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Screen_Shot_2022-10-09_at_9.05.03_PM.png

worldly kayak Oct 10, 2022, 2:05 AM

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okay yea it's the radius then

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mb

ruby juniper Oct 10, 2022, 2:05 AM

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Wait is what we came up here wrong?

worldly kayak Oct 10, 2022, 2:05 AM

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no everything we said before is right

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that's all good

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just, if I ever said "circle k" replace that mentally in your head with "circle with radius k" lol

ruby juniper Oct 10, 2022, 2:11 AM

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I'm still confused on what the moreover part says. So you invert points on a circle p (with center c) to a straight line

But there's no theorem from part (a-d) that says you can invert a circle to a straight line unless the straight line doesn't contain C??

worldly kayak Oct 10, 2022, 2:17 AM

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the center of p is a different point than C

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the "moreover" part is referencing case c

ruby juniper Oct 10, 2022, 2:19 AM

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So when we invert points on a circle p with a center c, we create a straight line not containing c (or vice versa-.. what's this)?

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This is what I understand so far

worldly kayak Oct 10, 2022, 2:19 AM

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the circle p does not have the center C

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rather, C is on the circle

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vice versa means the other way around, so case b

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C is the center of the original circle, not circle p

ruby juniper Oct 10, 2022, 2:23 AM

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So when we have a circle p and invert points on it, we get a straight line not containing c (or vice versa, which is case b) (also C is the center of some other circle not p)

This is what's basically going on

worldly kayak Oct 10, 2022, 2:24 AM

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yea

worldly kayak Oct 10, 2022, 2:24 AM

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ruby juniper So if I invert a point P on the blue circle that goes through the center of the ...

like in this example, C is the center of the purple circle

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and the blue circle is a circle p containing C

ruby juniper Oct 10, 2022, 2:25 AM

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So what's this "the line joining C to the center of p is perpendicular to m"

Like is there a line that goes through the center of c and p and is perpendicular to some other line m?

worldly kayak Oct 10, 2022, 2:26 AM

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so I think m is the line you get after inverting the circle

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and then the line going between C and the center of p is perpendicular to m

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here is a diagram

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hopefully this explains it better lol

ruby juniper Oct 10, 2022, 2:29 AM

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I thought we're doing case C where where inverting points on circles through C result in a straight line not containing C

worldly kayak Oct 10, 2022, 2:30 AM

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yeah so p is the circle we are inverting (notice that it goes through C)

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and it turns into m (a straight line not containing C)

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and now notice that the "moreover" comment holds, we have two perpendicular lines

ruby juniper Oct 10, 2022, 2:31 AM

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I'm confused where the second circle comes from... C is the center

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C isn't a circle

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worldly kayak Oct 10, 2022, 2:32 AM

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yeah C is the center of the green circle

ruby juniper Oct 10, 2022, 2:32 AM

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C is another circle though?

worldly kayak Oct 10, 2022, 2:32 AM

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no, C is the center of the original circle

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and then p is a circle that we are inverting

ruby juniper Oct 10, 2022, 2:33 AM

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But you drew a circle here? I'm probably not understanding something

Screen_Shot_2022-10-09_at_9.32.41_PM.png

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You have a circle labeled C

worldly kayak Oct 10, 2022, 2:34 AM

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oh C is labeling the center of that circle

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do you see the little green dot in the center of the circle

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that's C

ruby juniper Oct 10, 2022, 2:35 AM

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I think you're trying to show the center C by circling it in?

worldly kayak Oct 10, 2022, 2:37 AM

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C is the little dot, and I drew a circle of radius k around C

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and we are inverting the circle p

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and the circle p, after inversion, becomes the line m

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hopefully this didn't make it more confusing 😢

ruby juniper Oct 10, 2022, 2:42 AM

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worldly kayak hopefully this didn't make it more confusing 😢

I honestly think it would be better to do look over some examples off the theorem because even though I understand a-d word for word (a bit confused on the moreover) part, I don't know how to approach a question based off the theorem

Like here we want to find an inversion that transforms the circle x^2+y^2=16 to x=4... I honestly don't know which part of the theorem to use here

Screen_Shot_2022-10-09_at_9.40.30_PM.png

worldly kayak Oct 10, 2022, 2:43 AM

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kk

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so notice that you have a circle getting mapped to a line, which case of the theorem is that?

ruby juniper Oct 10, 2022, 2:49 AM

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worldly kayak so notice that you have a circle getting mapped to a line, which case of the the...

I honestly think I'm not understanding the theorem tbh... like for a I said inverting a point on a straight line that goes through C will end up resulting a point on that same line... You said it results in the "whole line"

worldly kayak Oct 10, 2022, 2:50 AM

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the idea for a is that you don't just invert one point on your line, you invert all of them

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and then what you end up with after you invert all of them is your original line again

ruby juniper Oct 10, 2022, 2:56 AM

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Ok so I came up with formal definitions of parts a-d

ruby juniper Oct 10, 2022, 2:56 AM

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worldly kayak the idea for a is that you don't just invert one point on your line, you invert ...

(a). Inverting points on a straight line containing C will result in the point ending up back on the line

(b.) Inverting points on a straight line NOT containing C will result into a circle that goes through C

(c.) Inverting points on a circle through C will result into a straight line not containing C

(d). Inverting points on a circle not through C will result into another circle that doesn't go through C

*Note that c and d are about a circle inside another circle

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Is this correct?

worldly kayak Oct 10, 2022, 2:56 AM

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wait how is this different from the original theorem

ruby juniper Oct 10, 2022, 2:56 AM

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I just explained it in my own words

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Like a way for me to understand

worldly kayak Oct 10, 2022, 2:56 AM

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oh okay

ruby juniper Oct 10, 2022, 2:57 AM

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Is it the same?

worldly kayak Oct 10, 2022, 2:57 AM

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that's correct, but your note isn't true, the theorem applies to any time where you want to invert a circle

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so the circle doesn't necessarily have to be inside your original circle

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it can be outside your circle

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or half inside and half outside

ruby juniper Oct 10, 2022, 2:57 AM

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inside or outside another circle*

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Can it be fully inside or outside?

worldly kayak Oct 10, 2022, 2:57 AM

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yes

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any type of circle, basically

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for part c, the circle goes through C, and for part d, the circle does not go through C

ruby juniper Oct 10, 2022, 2:58 AM

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*Note that c and d are about a circle being fully or half inside a circle (I changed it to this. Is that correct?)

worldly kayak Oct 10, 2022, 2:58 AM

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I don't think a note is necessary

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because the circle can be any circle

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it doesn't have to satisfy any conditions

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it just has to be a circle

ruby juniper Oct 10, 2022, 2:59 AM

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I think it should just be a circle engaging with a circle

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The note helps me

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*Note that c and d are about a circle engaging with another circle

So like this

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Also I would think about using part c of the theorem because we want to take points on a circle to a line (and that line doesn't contain C, as we can see)

worldly kayak Oct 10, 2022, 3:00 AM

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yeah like any of these situations are possible

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in the third case, the circle would invert to a line

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in the other cases, it would invert to another circle which does not contain C

ruby juniper Oct 10, 2022, 3:02 AM

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So we should essentially draw a circle engaging with C

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Wait

worldly kayak Oct 10, 2022, 3:03 AM

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well basically we have one circle with center C and radius k

ruby juniper Oct 10, 2022, 3:03 AM

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Ok so just wondering... how does the third case invert a circle to a line... and not the other ones?

worldly kayak Oct 10, 2022, 3:03 AM

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ruby juniper Ok so just wondering... how does the third case invert a circle to a line... and...

because of the theorem parts c and d

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so if the circle contains C, it inverts to a line

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and if the circle does not contain C, it inverts to a circle

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one way to understand why is that you can think of a line as a circle that goes through the "point at infinity"

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and then the center C will always map to the point at infinity

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and the point at infinity maps to the center C

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which is why a line (which always contains the "point at infinity") will always go through C after you do the inversion

ruby juniper Oct 10, 2022, 3:05 AM

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worldly kayak so if the circle contains C, it inverts to a line

So we need to find a circle that inverts points on the circle to a straight line (so we use part c)... Confused where you get part D from

worldly kayak Oct 10, 2022, 3:05 AM

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and then a curve that goes through C will invert to a line

worldly kayak Oct 10, 2022, 3:05 AM

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ruby juniper So we need to find a circle that inverts points on the circle to a straight line...

d is for the other situations I posted

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so situations 1, 2, and 4

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all of those circles map to other circles

ruby juniper Oct 10, 2022, 3:07 AM

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Ok so wait

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Lemme collect my thoughts

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So just wondering, what does it mean for a circle to be "through c" like does the circle have to have a point which is equal to the center of our original circle?

worldly kayak Oct 10, 2022, 3:10 AM

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oh so the circle being through C means that C is one of the points that the circle goes through, so it looks like this

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ruby juniper Oct 10, 2022, 3:11 AM

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so C isn't the center?

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I thought the C in Ick was the center

worldly kayak Oct 10, 2022, 3:12 AM

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C is the center of a circle

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but not the center of the circle that we're transforming

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it's the center of the circle that stays fixed after the transformation

ruby juniper Oct 10, 2022, 3:13 AM

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Isn't the center (0,0) here?

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Our new circle doesn't go through that

worldly kayak Oct 10, 2022, 3:13 AM

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no, C = (c,0) is the center

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for that question

ruby juniper Oct 10, 2022, 3:14 AM

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OH

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wait

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wait wait wait

worldly kayak Oct 10, 2022, 3:14 AM

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we want the circle to go through (c,0)

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so that it maps to a line

ruby juniper Oct 10, 2022, 3:15 AM

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So if we want to invert points on the circle x^2+y^2=16, we find a circle that goes through that circle (x^2+y^2=16 in this case) so we can apply part c which lets us invert points on that circle to help create a line that's not going through (-4,0)

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Wait

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I worded that poorly

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So if we want to invert points on x^2+y^2=16, we find a circle that has (-4,0) as a center to apply part c which lets us invert points on a circle with a center, say C (in this case -4,0) to create a line that doesn't go through (-4,0)

worldly kayak Oct 10, 2022, 3:19 AM

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yeah exactly!

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and this is also a good example as to why the "moreover" part is useful; let's imagine that you didn't know that the y-coordinate of C was zero

ruby juniper Oct 10, 2022, 3:20 AM

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What about (4,0), (0,4) and (0,-4)

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Why are they not an option

worldly kayak Oct 10, 2022, 3:20 AM

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you can actually figure out that the y-coordinate has to be 0 based on the "moreover" part of the theorem

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okay well first let's think about (4,0)

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based on the theorem, the circle gets mapped onto a straight line not containing C right?

ruby juniper Oct 10, 2022, 3:20 AM

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Oh yea

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Got it

worldly kayak Oct 10, 2022, 3:20 AM

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but if C = (4,0) then the line would contain C, so it's impossible

ruby juniper Oct 10, 2022, 3:21 AM

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so (0,4) and (0,-4)

worldly kayak Oct 10, 2022, 3:21 AM

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and the other two are impossible because they give you the hint that C = (c,0) so the y-coordinate of C is 0

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but it's actually possible to figure out that the y-coordinate of C has to be 0 without that hint

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note that the moreover part says that the line joining C and (0,0) (the center of the circle being inverted) has to be perpendicular to x=4

ruby juniper Oct 10, 2022, 3:22 AM

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Well C=(c,0) isn't included in the original problem

worldly kayak Oct 10, 2022, 3:22 AM

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therefore, the line joining C and (0,0) has to be horizontal

ruby juniper Oct 10, 2022, 3:22 AM

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That was his worked out solution

worldly kayak Oct 10, 2022, 3:22 AM

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oh alright lol

worldly kayak Oct 10, 2022, 3:22 AM

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worldly kayak note that the *moreover* part says that the line joining C and (0,0) (the center...

then this is why

ruby juniper Oct 10, 2022, 3:25 AM

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I think it's a good time for me to now understand the moreover part:

So when a circle goes to a straight line (or vice-versa) then the line joining the center of the circle C to the center of the circle P is perpendicular to a line m

(so when they both meet at their center, there's a line perpendicular to it?)

And when it's a circle to circle, their centers are C and colinear (what?)

worldly kayak Oct 10, 2022, 3:26 AM

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okay so

worldly kayak Oct 10, 2022, 3:26 AM

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ruby juniper Isn't the center (0,0) here?

let's take this example first

ruby juniper Oct 10, 2022, 3:26 AM

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Mhm

worldly kayak Oct 10, 2022, 3:26 AM

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so there's the point C which is (-4, 0)

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and then there's the center of the circle P which is (0, 0)

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and then there's the line m which is x=4

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do you get that so far

ruby juniper Oct 10, 2022, 3:27 AM

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Yes

worldly kayak Oct 10, 2022, 3:27 AM

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okay cool

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so all the "moreover" part is saying in this scenario is

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the line that goes between (-4,0) and (0,0) is perpendicular to the line x=4

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does that make sense?

ruby juniper Oct 10, 2022, 3:29 AM

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Like this?

Screen_Shot_2022-10-09_at_10.28.31_PM.png

worldly kayak Oct 10, 2022, 3:29 AM

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the vertical line should be x=4

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but otherwise yea that's right

ruby juniper Oct 10, 2022, 3:30 AM

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You said "the line that goes between (-4,0) and (0,0)" so I drew a line that goes through them

worldly kayak Oct 10, 2022, 3:30 AM

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yea that line is correct

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the horizontal line is correct

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but the vertical line should be at x=4

ruby juniper Oct 10, 2022, 3:32 AM

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I'm confused? Are you saying we need to draw a line that goes through (-4,0) and (0,0)? because if so I would've thought about the x-axis just

worldly kayak Oct 10, 2022, 3:33 AM

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yeah the line that goes through (-4, 0) and (0,0) is the x-axis

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and then notice how it's perpendicular to the line x=4

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which is the vertical line on the far right

ruby juniper Oct 10, 2022, 3:35 AM

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So like this? and if so, yea, I notice

Screen_Shot_2022-10-09_at_10.34.30_PM.png

worldly kayak Oct 10, 2022, 3:38 AM

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yea perfect!

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that's exactly what the "moreover" part is saying

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so we can use this information to conclude that C has to be on the x-axis

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and so C has to be (-4,0)

ruby juniper Oct 10, 2022, 3:45 AM

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worldly kayak yea perfect!

So when you take a circle, say p, onto a straight line (or vice versa) and there is a line going through the center of C (a different circle than p) and through the center of the circle p, that line is perpendicular to some other line, say m

Basically that's what we're saying

Basically B and C follow that condition^

worldly kayak Oct 10, 2022, 3:45 AM

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ruby juniper So when you take a circle, say p, onto a straight line (or vice versa) and there...

D doesn't follow that condition, but the rest of your explanation is perfectly correct

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B and C follow that condition

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A and D follow the second part of the "moreover" statement

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yea

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and just to add some details,

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if p is a circle, and I_C,k(p) is a line m, then the line going through C and the center of p is perpendicular to m

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that's for C

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the case for B is the exact same but flipped

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so if m is a line, and I_C,k(m) is a circle p, then the line going through C and the center of p is perpendicular to m

ruby juniper Oct 10, 2022, 3:48 AM

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worldly kayak so if m is a line, and I_C,k(m) is a circle p, then the line going through C and...

Ok so I need to rush this small discussion post real quick, but I'll be back in like an hour. Small question, but like how long will you be on tonight? I'm desperately trying to catch up on some lectures

worldly kayak Oct 10, 2022, 3:51 AM

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I probably won't be able to stay much longer

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unfortunately

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but hopefully what we've done so far was helpful

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this thread will probably close, but if you have other questions you can open another thread and someone else will help you :)

ruby juniper Oct 10, 2022, 3:57 AM

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worldly kayak this thread will probably close, but if you have other questions you can open an...

I got help here from someone before on a practice problem for a quiz and the same exact problem showed up on my actual quiz and I used the same approach and got it wrong 💀 I fear I'll get the wrong information if someone else helps

worldly kayak Oct 10, 2022, 3:58 AM

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rippp

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okay you can ping me when you have other questions, but I might not be around to answer

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all the helpers here are volunteers so unfortunately I can't invest a ton of time here haha but I enjoy helping when I can

ruby juniper Oct 10, 2022, 3:59 AM

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worldly kayak okay you can ping me when you have other questions, but I might not be around to...

Long shot but if you could look at these (review problems) questions briefly and provide me hints of what part of the theorem they used, I would appreciate it. Don't need no step by step, just would appreciate an idea since u might be off soon.

📎 Quiz_2_Review.pdf

worldly kayak Oct 10, 2022, 3:59 AM

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okay!

ruby juniper Oct 10, 2022, 3:59 AM

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worldly kayak all the helpers here are volunteers so unfortunately I can't invest a ton of tim...

True. And I appreciate your help

worldly kayak Oct 10, 2022, 3:59 AM

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no problem :)

ruby juniper Oct 10, 2022, 3:59 AM

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The answers are below btw

worldly kayak Oct 10, 2022, 4:01 AM

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I haven't looked carefully, so apologies if there any errors here

1a) part c and the "moreover"
b) no (why? use parts a and b of the theorem)
c) part d and the "moreover"

2a) part b of the theorem
b) part d of the theorem

part d of the theorem and the "moreover"
part c of the theorem and the "moreover"

ruby juniper Oct 10, 2022, 4:35 AM

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worldly kayak I haven't looked carefully, so apologies if there any errors here 1a) part c an...

Alright I'm back. So I wanna understand the second part of the moreover part... which is for A and B, right?

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Especially since a lot of these questions use that part of the theorem

worldly kayak Oct 10, 2022, 4:35 AM

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it's for a and d

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okay I can't stay for long but I can draw a picture and hopefully it should be self-explanatory

ruby juniper Oct 10, 2022, 4:36 AM

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"hopefully", for me 😅

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Go for it

worldly kayak Oct 10, 2022, 4:37 AM

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okay let's say you have some point C once again

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and you have another circle and it's centered at B

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and you calculate the inversion of the circle and end up with the blue circle

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which is centered at A

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then, A, B, and C are colinear, i.e. they all lie on the same line

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that's all it's saying

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oh I just realized I made a typo earlier when I said it applied to a and d

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it only applies to d lol

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anyways hopefully this helps

ruby juniper Oct 10, 2022, 4:38 AM

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and you calculate the inversion of the circle and end up with the blue circle

Inversion of which circle?

worldly kayak Oct 10, 2022, 4:38 AM

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inversion of the purple circle

ruby juniper Oct 10, 2022, 4:38 AM

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Yea i was thinking D since it's circle to circle

worldly kayak Oct 10, 2022, 4:39 AM

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so I_C,k(the purple circle) = the blue circle

ruby juniper Oct 10, 2022, 4:39 AM

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And you mean like inverting all the points on the purple circle gets us the blue circle?

worldly kayak Oct 10, 2022, 4:39 AM

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yup

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okay gotta go but hopefully this helped!

ruby juniper Oct 10, 2022, 4:41 AM

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Darn pensivebread

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--close

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Ugh how do I close this

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Found it

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I'll create a new discussion

worldly kayak Oct 10, 2022, 4:45 AM

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.close

quartz folioBOT Oct 10, 2022, 4:45 AM

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Solved

Post marked as solved by @worldly kayak.

Use .unsolved if this was a mistake.

ruby juniper Oct 10, 2022, 4:45 AM

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worldly kayak .close

Is this an archive for me?

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It's still up

worldly kayak Oct 10, 2022, 4:47 AM

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it archives eventually, but you'll still be able to see it regardless

ruby juniper Oct 10, 2022, 4:50 AM

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worldly kayak it archives eventually, but you'll still be able to see it regardless

If your still on, quick question, does the moreover for case C (and vice-versa B) stay the same?

worldly kayak Oct 10, 2022, 2:44 PM

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worldly kayak if p is a circle, and I_C,k(p) is a line m, then the line going through C and th...

yes it's this ^

ruby juniper Oct 10, 2022, 2:47 PM

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worldly kayak yes it's this ^

Hey, are you available to help?

worldly kayak Oct 10, 2022, 2:47 PM

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not right now sorry

ruby juniper Oct 10, 2022, 2:47 PM

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Darn 😔 When are you usually available? I want to be considerate

worldly kayak Oct 10, 2022, 2:50 PM

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probably not anytime soon, sorry

#Anyone understand inversions?