Vincent Pantaloni
I have been thinking of this...
Every triangle has an incircle and a circumcircle. Given a (blue) circle and a (pink) circle inside the blue one, is there always a triangle for which these are the circum and inner circles ? If not, then when ? How to find one, some, all ?
#Thread 🧶
Nov 1, 2022 at 1:22pmWeb
Show threadVincent PantaloniNov 1, 2022
For the first question "is it always possible to find a triangle..." the anwer is clearly "No.".
My daughter pointed out that if both have the same center, the triangle can only be equilateral, so if the radius is not right you won't find a triangle.
Show threadVincent PantaloniNov 1, 2022
Say the blue circle has a unit radius. The equilateral triangle gives the greatest incircle possible with a radius of 1/2. So any radius r>1/2 won't yield a solution. So let's say I choose a radius r<1/2 for the potential incircle, inside the blue unit circumcircle. Can we always find a triangle ABC ? If not then when and how ?
Show threadVincent PantaloniNov 1, 2022
First some exploration. For given points A and B on the circumcircle, and C going around the circle, where are the incircles of ABC ?
Show threadVincent PantaloniNov 1, 2022
What about the centers of the incircles, where are they ? Consider the points D and F for which CA=CB and draw circular arcs AB from centers D and F.
#thread 5/n
Show threadVincent PantaloniNov 1, 2022
Now let's move B around and draw all the positions of the incircles and their centers for triangles ABC incribed in the blue circle.
#thread 6/n
Show threadVincent PantaloniNov 1, 2022
Incidently we get a lovely figure :)
#MathGIF #MathArt
#thread 7/n
Show threadVincent PantaloniNov 1, 2022
No, back to our problem. For a given radius r<0.5 for the potential incircle can we find a triangle ABC ? Yes, but the pink (in)circle can't be anywhere in the blue circumcircle.
#thread 8/n
Show threadVincent PantaloniNov 1, 2022
Given radius r<0.5 we can look for a triangle with a 60° angle which has such a circle as incircle since an equilateral triangle gives the max incircle (radius=0.5).
Here's an example for a radius r=0.3. For a given vertices A & B such that <AOB=120° we get two positions for in the in-center I. That we can determine what we saw earlier.
#thread 9/n
Show threadVincent PantaloniNov 1, 2022
By drawing the arc with center N going through A and C we get the two possible positions for the incenter (for a 60° angle in B).
#thread 10/n
Show threadVincent PantaloniNov 1, 2022
Now the amazing thing ! So you found one specific position for this incircle for a triangle with a 60° angle but what about other triangles with this incircle ?
Pick any point A on the blue circle, draw the tangents to the pink circle and you have a triangle which has the pink circle as incircle and the blue as circumcircle.
This a version of Poncelet's closure theorem for three points. #MathGIF.
https://en.wikipedia.org/wiki/Poncelet%27s_closure_theorem
#thread 11/n
Poncelet's closure theorem - Wikipedia

Show threadPig AnkleNov 7, 2022
@panlepan very satisfying result. Your illustrations are also very satisfying. Can I ask what you use to make them?
Show threadJohn Carlos BaezNov 1, 2022

@panlepan - wow, great stuff!!!

One little tip: if you're writing a series of toots here, for all but the first one you should click on the little globe that says "Change post privacy" when you mouse over it, and change the toot to "unlisted". This means we can still see it, but it won't be announced in our Notifications. If you don't do this, we see all the toots in your series - but in reverse chronological order, which can be confusing.

Show threadVincent PantaloniNov 1, 2022
@johncarlosbaez Thanks for the tip ! I'll continue like this. It's my first thread here 😬
Show threadColin the MathmoNov 1, 2022

@panlepan There's no need to pnic or be self-conscious about it ... we are all learning about "best practice"

@johncarlosbaez

Show threadOscar CunninghamNov 1, 2022
@ColinTheMathmo @panlepan @johncarlosbaez I have a question about this. I selected 'Unlisted' for this https://mathstodon.xyz/@OscarCunningham/109268407653365207 post. But I still saw it in my 'Home' feed. Did everyone else see it in their feeds? I thought the point of unlisted was to stop that.
Oscar Cunningham (@[email protected])

@seano @BartoszMilewski @MartinEscardo @yog In fact, could we say that \(¬¬A\) implies \(A\) if and only if \(A\) is equivalent to \(¬B\) for some \(B\)?

Mathstodon
Show threadColin the MathmoNov 1, 2022

@OscarCunningham "Unlisted" still means visible for everyone, but it's "opted-out" of discovery features. So people on the instance probably still see it, whereas people *not* on the instance probably only see it if they follow you, or click on the thread.

(This answer is not definitive, and may be wrong)

@panlepan @johncarlosbaez

Show threadOscar CunninghamNov 1, 2022
@ColinTheMathmo @panlepan @johncarlosbaez That makes sense. I saw all of John's last thread in my feed even though he unlisted all but the first post.
Show threadColin the MathmoNov 1, 2022

@OscarCunningham Beware ... I'm the master of appearing plausible when I am in fact wrong.

@panlepan @johncarlosbaez

Show threadOscar CunninghamNov 1, 2022
@ColinTheMathmo @panlepan @johncarlosbaez Aha! I found the full explanation of what 'Unlisted' does. https://docs.joinmastodon.org/user/posting/ It appears in people's 'Home' feeds but not in the 'Local Timeline' or 'Federated Timeline'. Which is a shame because what I'd really want is posts which are visible to everyone, but which don't polute people's Home feeds.
Posting to your profile - Mastodon documentation

Sharing your thoughts has never been more convenient.

Show threadJohn Carlos BaezNov 1, 2022

"I saw all of John's last thread in my feed even though he unlisted all but the first post."

Did you see it all in your feed in reverse chronological order - the most recent ones on top? That's the effect I'd like to avoid.

Show threadOscar CunninghamNov 1, 2022
@johncarlosbaez Yes, it looks like this.
Show threadJohn Carlos BaezNov 1, 2022
@OscarCunningham @ColinTheMathmo @panlepan - I see all my 'unlisted' posts under my profile, in reverse chronological order. But in other contexts the unlisted ones do not appear in the feed, though they're still easily reachable.
Show threadIanagolNov 1, 2022
@panlepan Nice construction! This follows from the fact that the incenter is the intersection of the angle bisectors, and inscribed angle angle theorem.
Show thread⎯ΘωΘ⟶Nov 1, 2022
@panlepan the generalization of this class of problems to higher dimensions and exotic geometries is a profound well of treasures
Show threadParthNov 1, 2022
@yog @panlepan any book or something for this treasure?
Show thread⎯ΘωΘ⟶Nov 1, 2022
@XinYaanZyoy @panlepan i would suggest the recent masterpiece "Geometric Regular Polytopes" by McMullen, and furnish with healthy doses of differential cohomology (let's cite Bott and Tu)
Show threadVincent PantaloniNov 1, 2022
@yog @XinYaanZyoy Thanks for the refs. Right now I'm experimenting on my own and I realize there's more than I expected...
Show threadSmittyHalibutNov 1, 2022

@panlepan it’s pretty easy to think of an example of two circles that don’t have a triangle: the inner pink circle is just a little bit smaller than the larger blue circle. So, not every pair of circles.

I can’t think of how to determine if there is a triangle though.

Show threadVincent PantaloniNov 1, 2022
@smitty Right ! That was also the first counter-example that popped in my mind. My daughter, though, thought of an other extreme example, a tiny circle centered at O. Check out the rest of the thread (WIP). :)
Show threadSmittyHalibutNov 1, 2022
@panlepan another example is when the inner circle touches the outer circle. There’s no way to get a straight line through that gap without it being tangent to both circles.
Show thread0xDENov 1, 2022
@panlepan There isn't always a triangle, but when there is one there are infinitely many. You can start at any point on the outer triangle, find tangent lines to the inner circle, follow them around back to the outer circle, etc. It also works for polygons with more sides than triangles — you always get another polygon with the same number of sides! See https://en.wikipedia.org/wiki/Poncelet%27s_closure_theorem
Poncelet's closure theorem - Wikipedia

Show threadVincent PantaloniNov 1, 2022
@11011110 Yes, Poncelet's theorem is the amazing result on this topic. I'll mention it soon in the thread. :)
Show threadfeuerfranNov 1, 2022
@panlepan Seems interesting, but I don't understand.
Show threadCaiNov 1, 2022
@panlepan gorgeous visualisations!
Show threadVincent PantaloniNov 1, 2022
@cai Thanks ☺️
Show threadSylvainNov 6, 2022
@panlepan I wouldn't be surprised if from a projective geometric viewpoint, it required the existence of a Möbius transformation preserving the infinite point and mapping the vertices of the desired triangle to the intersection points of its sides and the inner circle.