Triangles have a Magic Highway - Numberphile

[@Kryoclasm]

You should have Zvezdelina do more videos, I never get bored when she is explaining something.

[@CastorQuinn]

I cannot get my head around how Zvezdelina can draw all these diagrams so well just by hand. I can't even manage a straight line by hand at all, let alone one which bisects an angle and meets a line at its midpoint.

[@timetogetcancer7866]

And then you see them draw a 3d shape

[@NikopolAU]

+Castor Quinn quit drinking then

[@yali_shanda]

In Soviet Russia, triangles draw you.

[@yali_shanda]

+Yali Shanda Or, should I say in this case, Soviet Bulgaria.

[@bgezal]

+Castor Quinn Just draw triangles for a couple of decades and you will also become master.

[@Vank4o]

1:00 Nice nod to the Vitosha computer, the first Bulgarian made computer :)

[@vailias]

+Scrotie McBoogerball Thank you! I could read the text but didn't know the word. (google translate was of zero help also)

[@SparklyRazor]

+Scrotie McBoogerball Ah, I wondered what that was!

[@ivayloivggrigorov9959]

+vailias it's also called after a mountain.

[@rdreher7380]

+Scrotie McBoogerball Ah that's what it was. I know Russian, so I could read it, and I figured out it was Bulgarian, and the name of a mountain, but I had no idea what the reference was here.

[@icyzoneinfo]

I was just going to ask what does the mountain have to do with computers :D

[@MikeDawson1]

We've learned two things: - the animations are VERY well done - that lady REALLY likes triangles :)

[@malinastankova4918]

lol

[@Triantalex]

false.

[@IaKhanic]

Damn you got Question 6 Right!!!

[@unounk9415]

in less than 4.5 hours!

[@unounk9415]

it took the guy in the main video a YEAR to solve it

[@unounk9415]

in less than 4.5 hours! It took the guy in the main video a YEAR to solve it, and the hosts of the competition couldn't solve it in 6 hours

[@sobanudlz]

Im still Uno Unk

[@hawthornroot]

with a perf score of 7

[@xmotoFF]

“Some of his [Euler's] simplest discoveries are of such a nature that one can well imagine the ghost of Euclid saying, 'Why on earth didn't I think of that?'” H. S. M. Coxeter

[@Triantalex]

??

[@barmansushi]

Pete, nice work on the animations, really helps with visualisation

[@pmcpartlan]

+Tom D.H Thank you, glad they helped.

[@smaakjeks]

+Pete McPartlan Yeah, great job!

[@AD173]

+Pete McPartlan Hey, what software do you use for the illustrations? I really need to know!

[@xnopyt647]

@@pmcpartlan You are awesome!

[@ErikOosterwal]

At one point the animations looked like 3D representations with the triangle and medicenter lying on a plane and the circumcenter and orthocenter positioned above and below the plane. In this simulated 3D view it looks like the Euler line is perpendicular to the plane. 🤔

[@KalebPeters99]

Wow, I love when such simple geometry can produce such a seemingly magical result! And side-note; the graphics in this video were _awesomely_ done.

[@BlueGatorade0]

on which point of a triangle is the hospital located? the medicenter!

[@laxpors]

+Kristian Bernardo HA

[@SpaghettiFace2]

I would make a similar joke about the circumcenter, but it would just be awkward.

[@FernieCanto]

+SpaghettiFace2 I tried to do a circumcenter joke too, but it was cut.

[@LosDynasty]

+Fernie Canto I would make a joke about the orthocenter but it's not funny. its unorthodox. ( i tried. bye)

[@EvolBob1]

+Kristian Bernardo- Its funny. Its even funnier telling this, especially when I get a blank stare and I'm the only one laughing.Explaining it only makes it worst.

[@ba_livernes]

The "Nah just kidding" at 4:00 killed me

[@judahdelrio3650]

FliiFe what

[@jamesterpaul]

Judah Del Rio ahlie

[@sobanudlz]

More like 4:11 ALSO WATCH MY CHANNEL

[@screamsinrussian5773]

@@sobanudlz no, go away

[@jassenjj]

Did you think of Kristen Wiig? Just kiddin'...

[@yellowmeerkat97]

I love the videos with helpful animations from Pete McPartlan and I love the videos with Zvezdelina Stankova, so this is absolutely wonderful. Thank you for the gift, Brady.

[@AlekVen]

7:13 This really does look like a rotation in 3D rather than some purely 2D transformations. Cool.

[@ErikOosterwal]

In this perspective it looks like the Euler line is perpendicular to the plane containing the triangle and medicenter.

[@dougmercer]

This is one of my favorite numberphiles to date. A charming result, presenter, and animations.

[@regulargold7065]

The beauty of the Euler line is that it means there is a triangle around every line

[@jkid1134]

Very solid and rigorous proof there, dancing a triangle about graphically

[@meri7108]

I really love the way Zvezdelina explains things!

[@jordantistetube]

"Ooh! Fancy. I can get wild! Oo-ho!"

[@JLConawayII]

The medicenter is where I have to go after watching this. My head hurts.

[@Triantalex]

??

[@ricardo.mazeto]

These videos makes me fall in love with maths!

[@tylerborgard8805]

I just thought of 4 new centers for a triangle, using the 4 that were introduced in this video. I haven't thought them through that much, but I'm interested in seeing if there are any weird mathematical properties about these centers. So here we go: 1. Anti-orthocenter: Take the centroid, circumcenter, and incenter of any triangle (that is, all the centers except the orthocenter), and those points will form a new triangle. Repeat the process for the new triangle, and for the next triangle, etc. Hopefully, the triangles should get progressively smaller and converge to a point. That point is the anti-orthocenter. 2. Anti-centroid: Go through the same process that you would to find the anti-orthocenter, but this time use the circumcenter, incenter, and orthocenter (that is, all the centers except the centroid) as your three triangle-forming centers. 3. Anti-circumcenter: Same process as the previous two centers, but this time use the centroid, orthocenter, and incenter (that is, all the centers except the circumcenter) as your three triangle-forming centers. 4. Supercenter: Take the previous three centers of any triangle, and they will form a new triangle. (Actually, I have no idea if they do. It could be the case that the anti-orthocenter, anti-centroid, and anti-circumcenter are always collinear for all I know. That's an open question, and I'm interested in seeing a proof either way.) If they do form a triangle, take the anti-orthocenter, anti-centroid, and anti-circumcenter of that triangle to form another one. Repeat this process ad infinitum. Hopefully, these triangles will also get progressively smaller, and the point they converge to is the supercenter. Questions I'm interested in having answered: For which triangles do these centers exist, and for which triangles do they not? What I already know is that the center in question will not exist if one of the triangles along the way is actually a straight line (which is why there is no anti-incenter in this list), or if the triangles do not get smaller in a way that converge to a point. If the sequence of triangles constructed in calculating any of these centers doesn't converge to a point, what happens to them? Do any of these centers lie on the Euler line? If so, which ones? Is there a group of three of these centers that will always be collinear, provided they exist? Are there two centers (out of the ones I listed and the ones in the video) that are actually the same point in disguise? Are there any weird relationships between the smaller triangles constructed along the way and the original triangle? For example, are they similar? Do they share a common centroid, circumcenter, incenter, or orthocenter? How do the areas and side lengths compare?

[@non-inertialobserver946]

Very interesting

[@thiagovscoelho]

my favorite property of the centroid (in Portuguese it's the 'baricentro') is that it's the triangle's center of gravity. this means that a triangle can be balanced on that point

[@GenericInternetter]

thanks, capitao obvio

[@shambosaha9727]

Barycentre literally means Centre of Mass

[@shambosaha9727]

Also, the centroid is the barycentre of just a triangular plate. The barycentre of a triangle-shaped wire is the Spieker centre.

[@OptimusPhillip]

*assuming that the weight distribution across the area of the triangle is constant.

[@JDSileo]

I could listen to Professor Stankova lecture all day.

[@pleonov]

greetings from Bulgaria! Great video Zvezdelina amd Brady!

[@Henrix1998]

I really like her accent

[@spaceminers]

This can explain metaphysics, quantum physics, faster than light travel as well as help solve the three body problem

[@themobiusfunction]

3:44 except when you are dealing with an equaliteral triangle

[@TheAAMoy]

This was figured out how LONG ago, and people are still wowed by it. Cause Math and Science ROCK!

[@ExaltedDuck]

anyone else notice during the animations that the Euler line coincides with the 2d projection of a line orthogonal to the plane of the triangle through its centroid? Fascinating.

[@ExaltedDuck]

...that is if we perceive the triangle with fixed vertices and rotating in a 3 dimensional space and projecting onto the plane as well.

[@schnuffelwuff]

And the Circle is the 2D Representation of a Sphere

[@schnuffelwuff]

+Patrick Waldner Okay this one may be wrong

[@smaakjeks]

+ExaltedDuck Yep!

[@Mathhead2000]

I was about to comment the same thing. They should make a follow-up video on that.

[@NerdGlassGamingPA]

I am in love ! And I am not even a Mathmatician !!! This is awesome ! Ms. Stankova is also so awesome !

[@jrgmen]

Wow Brady! The editing and animation has really improved! Keep up the great work!!

[@puerto6482]

Витоша (pronounced vitosha) was the first Bulgarian computer built in 1962-1963 on the basis of a cultural agreement between the Romanian and Bulgarian academies of science.

[@rgalt5675]

This is arguably my favorite numberphile video. I love number theory but would to see more geometry, trigonometry, and calculus videos.

[@Regular-Sized]

"I can get wild" well that made my day

[@Latrocinium086]

That was some great and pertinent geometry animation. Excellent job! Thanks

[@andydaniels6363]

The first animation that shows the initial triangle being warped into others nicely illustrates how one triangle can be mapped onto another via an affine transformation. Since they preserve intersections, that’s a way to prove that the medians of any triangle are coincident.

[@apid4075]

The animation at 7:15 looks like as we had a equilateral triangle rotating in 3D space with a orthogonal line (perpendicular to a plane the triangle lies on) led trough the medicenter. So when all the centres collapse it's like we're looking at the triangle "from the top".

[@ThomasGodart]

Ah ah, beautiful! Everybody would probably enjoy to have a teacher like that, she's turning simple Maths facts into fascinating questions and wonders. Just like James Grime ;-)

[@s.d.s.7007]

That is elegant! I love to learn new concepts and see where they apply.

[@migfed]

Brilliant Zvezdelina and Brady. Geometry is such a nice discipline.

[@duckofdeathv1595]

Zvezdelina is awesome. Love her videos. Thanks Brady!

[@JackSwatman]

centroid wins for me, can't have a centre that lies outside of the shape.

[@justahker3988]

+JackSwatman Incentre also can't lie outside the shape.

[@tylerborgard8805]

+JackSwatman If the center can't be outside the shape, then what about the center of a donut?

[@Tumbolisu]

+Tyler Borgard Not fair, that's a concave object.

[@NotQuiteFirst]

rekt

[@JackSwatman]

+Tyler Borgard I don't feel that totally nullifies my statement but it was very clever and unarguably true

[@zaharimarinov4289]

My mother Joanna Stoicheva Ivanova knew Zvezdelina in the 7th grade. They were in the same Bulgarian school in Ruse. They both had maximum points on the final exam(and another boy). But now my mother is a psychology teacher with 400$ monthly salary (because Bulgaria corruption ect.) and Zvezdelina is having hundreds of thousands of views from America... Поздрави от България!

[@user-zb8tq5pr4x]

Zvezdelina is getting less from this video than your mum

[@hats1642]

For any triangle it is possible to construct a circle which passes through the midpoint of each edge, the foot of each altitude, and the midpoint of the line segment from each vertex to the orthocentre. The centre of this circle is called the nine-point centre, and it is another centre which lies on the triangle's Euler line.

[@toxicdesire8811]

yep, best handwriting I've seen on numberphile.

[@hovikghazaryan9130]

I'm so happy I found this, I'm learning it in school rn and I've been having trouble

[@jhoughjr1]

This is one of my favorite numberphile videos

[@terrygoyan]

I love the Numberphile videos! They get the most fascinating people in them Thank you!

[@rowdy35967]

Love the animations, well done!

[@PanozGTR2]

I like the centroid as it is the center of mass, however my favourite center is the nine-point center. It also lies on the Euler line, btw. It is the midpoint of the orthocenter and the circumcenter, although that isn't the definition.

[@lxathu]

Usually, I watch Np to hear interesting things not heard before. This time it was a time machine taking me back 25-30 years and it was gooood.

[@DynestiGTI]

My favourite Numberphile video.

[@NostalgiaGames_Gamer]

is it bad that i see the triangles and the lines as 3 dimensional ?

[@emilysofiadelatorremartin524]

i don't think so

[@Hilko26]

+Watchable No I had it too. It's just an automatic process of your brain trying to comprehend the things happening on the 2d screen.

[@moazzamak]

+Watchable It's worse then I expected. I'm afraid you have "The knack" :P

[@Satchboy71]

+Watchable When they moved the lines around it really did look three dimensional. The Euler line looked like the Z axis of sorts.

[@CryZe92]

+Watchable No, because once you have at least 4 points, a 3-dimensional projection can be clearly defined. So the 3 vertices of the triangle plus the additional center point form a 3-dimensional projection, making it look like it would be 3-dimensional.

[@РумянаСтанкова-с7ш]

Great presentation and great animation!!

[@fatalfruit2662]

I never realized that math in Bulgaria is taught differently than in any other country, even though that might seem obvious. Despite that, I never would have imagined that there was a relationship between all those different centers of a triangle. Great video and many thanks to Zvezdelina for the explanation. Поздрави!

[@mariebcfhs9491]

I love the equilateral triangle, it is the most beautiful and symmetric shape to me

[@venkybabu8140]

That's why half clusters are famous. Half is something to do with property of circle. Because radius is equal all through. Angles show for special properties. And circumcenter for inversion. Inversion can happen when you have equiangular. Just frequency match. Or Octavia.

[@rushabhshah8981]

zvezdelina stankova.... your handwriting is awesome

[@lawrencecalablaster568]

:D I loved learning about the different centres of a triangle in 9th grade geometry. Awesome!

[@wanderleyapparecidovieira2282]

Just now I've seen this video,congratulations for the perfect pronunciation !

[@sarsandtripe]

Zvezda is so good, I love her work

[@chentiangemalc]

really good video & animation ... excellent presentation from Zvezdelina Stankova, also excellent freehand diagram drawing skills

[@FernandoRodriguez-ge2tg]

My favorite video video in a while

[@errlshmirl3130]

My teachers did not show us how math could be applied to so many life problems. Even in high school I still didnt know that algebra describes 2d, 3d, and shapes. EVERYTHING. Better late than never

[@thomasolson7447]

My favorite video so far.

[@keensauce]

5:15 , love she has a favourite (and her explanation as well is so cool)

[@Triumvirate888]

Whoa... so you can literally represent a triangle in 1-dimensional space just by measuring the movement of dots along the line!? Amazing! I wonder if that exists for other shapes as well.

[@bradzepfan]

very very well done! very entertaining! i can't wait to show it to my daughters!

[@zero_spin]

There's one more centre called EXCENTRE .. where two external bisectors and one internal bisector of a triangle are concurrent . It holds a special property too : INCENTRE (corresponding to internal angle bisector) and EXCENTRE of a triangle are Harmonic Conjucates of each other ;)

[@randomusername3388]

1:14 ooh fancy I can get wild ooOoOoh

[@bbkandsons]

Brilliant explanations!

[@flux202]

Congratulations on question 6 ma'am👏👏

[@bulman07]

Weird how you see the moving triangle as 3D. Is there a name for that like pareidolia?

[@hatchetxrip]

The dissociation between vision-for-perception and vision-for-action

[@crackedemerald4930]

We are used to 3d space, if we see a 2d object in 2d space that resembles a projection, outline or structure of a 3d object, we are going to see a 3d object

[@TimeSynthis]

A center of buoyancy must higher than a center of gravity for an object to float. So different centers do have real world design implications. Interesting video, thanks.

[@powaybob45]

40 years ago I taught high school geometry for a few years before returning to grad school. I wish I had discovered the Euler line relationship to triangles to spice up the class for a day or two.

[@davidross3487]

Question: Given the three "centers" is it possible to determine the triangle that generated them? If not, what is the class of triangles that may have generated them? What is the situation in the degenerative cases where two or all three of the "centers" coincide? Any thoughts?

[@chunawalla]

Fantastic stuff, thoroughly enjoyed this!! One of the things I recently learnt while reviewing analytic geometry is the theorem of Ceva. The cevians - medians, altitudes and angle bisectors are concurrent.

[@frizider2]

I need a wife that will look at me like this woman looks at triangles.

[@syedwaleedshah2830]

well you gotta start drawing triangles on your body then , eh ?

[@rafciopranks3570]

What would be the sum of their angles?

[@nicxtrem21]

Imagine if she'd date food-writing glass-structure geometry genius guy

[@mienzillaz]

Any update?;)

[@fishsauce2221]

Was going to make a triangle joke but I didnt come up with anything.

[@alfiestoppani]

This was the best thing I have ever seen.

[@nmmm2000]

Nice touch with "Vitosha" on the computer :) My aunt worked on this computer back in 1961.

[@nickpancione5084]

I like the new style for the animations!

[@ComputerRouter]

I've watched this video before and wasn't too interested.... Just seen VSauce video about the tee-shirt in the new curiosity box, and now I'm totally engrossed by this video

[@Caye2013]

Amazing video! This woman is magical!

[@leas5365]

This is so awesome!

[@LunarFurorGames]

The line looks like it always runs perpendicular to an equilateral triangle directly from the center, when they move it around you can see it. and the direction of the 2d "highway" is just based on your 3d perspective.

[@Sylocat]

What about the Steiner Point(s?) of a triangle? For those who don't know, the Steiner Tree problem is to find the minimum combined lengths of straight line to connect all the vertices in a given pattern. So, if you have three vertices, A and B and C (like, say, in a triangle, with sides AB, BC and AC), then plant a vertex D and draw lines AD, BD and CD. Then add up the lengths of AD, BD and CD. Your goal is to find the point D where the sum AD + BD + CD is lowest. You'd think this would be the centroid, but it usually isn't. Of course, most versions of the Steiner Tree problem are NP-complete, so it might be tough to plot this.

[@mputuchimezie7966]

The technology's sound is killing me 😂 But ma'am you are fantastic and I enjoy learning from you.

[@WayneLinnlikestouseGeoGebra]

Numberphile never seize to amaze me.

[@mojosbigsticks]

Fascinating. Thank you.

[@overwhelmingsarcasm]

Another great video and a much enjoyed nod to my home country with the 'Витоша' computer ;) Браво!

[@Pika250]

The centroid has a 2:1 ratio of distances from orthocenter to centroid, to from circumcenter to centroid, as though the orthocenter was treated like a vertex and the circumcenter was treated like the opposite edge's midpoint.

[@KasabianFan44]

Is there a constant ratio between the distance from the centroid to the circumcentre and the distance from the centroid to the orthocentre? I've been looking at the animation for the past 20 minutes and I swear that this is the case...

[@danieleboccanfuso1723]

You have intuition bro, because there is a ratio. calling the centroid G the circumcentre O and the orthocentre H the ratio is OH/GO=3

[@KasabianFan44]

Daniele Boccanfuso I've just watched the extra footage, it's explained there :D I guess I should have watched it before commenting lol Thanks for the reply anyway :-)

[@danieleboccanfuso1723]

+KasabianFan44 You are welcome :)

[@Supuhstar]

loving the fancy graphics!

[@WahranRai]

the 3 centers H, C, O of Euler line verify : HC = 2 OC

[@gregorymccoy6797]

Very nice. Mind expanded.

[@PunmasterSTP]

Eulearned a ton of information from this video, and I hope to see Zvezdelina Stankova again!

[@dco901]

This is one of the many reasons triangles are the coolest.

[@yjawhar]

So, if I break my leg, do I go to the orthocenter or is it better to go to the medicenter for a full check up?

[@mayabartolabac]

Neither. You should go to the circumcenter because it's most likely the closest point to you.

[@huskytail]

1:00 Витоша 🤘 All of that returns me to Bulgarian middle and beginning of high school 😁😍

[@kennethflorek8532]

5:45 But, myself, after being impressed at what you could prove from very little, it seemed the proofs just about never explained why the magic happened. So we have 4 different cases of a coincidence, that can be proved separately, but no general reason for the magic. Often theorems about triangles can be generalized to more-sided figures, provided the figures can be inscribed in a circle. It just happens that a triangle can always be inscribed in a circle, but not so for more sides. But a generalization of any kind typically gives you a better idea of what is going on.

[@StevenForditude]

Fantastic. I'm having fond memories of my high school geometry class. 😊