Triangles have a Magic Highway - Numberphile

Triangles have multiple centres, and many of them lie on the so-called Euler Line.
More links & stuff in full description below ↓↓↓
Extra footage: • Triangle Centres and t...
Featuring Zvezdelina Stankova.
Editing and animation by Pete McPartlan.
Thanks also to Tom Davis of geometer.org
More videos with Zvezda: bit.ly/zvezda_videos
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Videos by Brady Haran
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[@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. 🤔

[@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.

[@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'...

[@piscesfool1596]

great animation!

[@numberphile]

+Pisces Fool kudos to Pete for that... he did some nice stuff.

[@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.

[@KrisKrisKrisKrisKris]

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.

[@JLConawayII]

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

[@Triantalex]

??

[@Imagine-Baggins]

Animations were top notch this episode

[@jordantistetube]

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

[@JDSileo]

I could listen to Professor Stankova lecture all day.

[@dougmercer]

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

[@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.

[@Regular-Sized]

"I can get wild" well that made my day

[@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.

[@ricardo.mazeto]

These videos makes me fall in love with maths!

[@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.

[@meri7108]

I really love the way Zvezdelina explains things!

[@Henrix1998]

I really like her accent

[@TheAAMoy]

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

[@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

[@jrgmen]

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

[@Latrocinium086]

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

[@pleonov]

greetings from Bulgaria! Great video Zvezdelina amd Brady!

[@rgalt5675]

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

[@randomusername3388]

1:14 ooh fancy I can get wild ooOoOoh

[@s.d.s.7007]

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

[@rowdy35967]

Love the animations, well done!

[@NerdGlassGamingPA]

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

[@terrygoyan3022]

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

[@migfed]

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

[@duckofdeathv1595]

Zvezdelina is awesome. Love her videos. Thanks Brady!

[@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.

[@ratlinggull2223]

So... Illuminati is Magical?

[@theshadowmonster1]

+Fiend Sweg Basically

[@cubedude76]

+Fiend Sweg is your icon the shade from warcraft 3?

[@EebstertheGreat]

+cubedude76 Yes, it's the beveled Undead Shade icon, which in DotA is used for the Shadow Fiend.

[@Cloiss_]

+Fiend Sweg Thats how our 8th grade Geometry class views it (even the teacher)

[@ratlinggull2223]

cubedude76 Yup, my icon is pretty much Shade with Shades.

[@user-wm7nk7db6p]

Great presentation and great animation!!

[@chentiangemalc]

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

[@jhoughjr1]

This is one of my favorite numberphile videos

[@leas5365]

This is so awesome!

[@thomasolson7447]

My favorite video so far.

[@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.

[@FernandoRodriguez-ge2tg]

My favorite video video in a while

[@fernandoschuindt1665]

This is fantastic!

[@hovikghazaryan9130]

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

[@mojosbigsticks]

Fascinating. Thank you.

[@nickpancione5084]

I like the new style for the animations!

[@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 ;-)

[@wanderleyapparecidovieira2282]

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

[@gregorymccoy6797]

Very nice. Mind expanded.

[@sarsandtripe]

Zvezda is so good, I love her work

[@bradzepfan]

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

[@overwhelmingsarcasm]

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

[@StevenForditude]

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

[@Dan-ms4oq]

This is an amazing video!

[@lawrencecalablaster568]

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

[@UltraJordanGaming]

Amazing animations!

[@Supuhstar]

loving the fancy graphics!

[@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.

[@dasibaho]

Beautiful.

[@Cyberial]

This is super cool!

[@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

[@aybmnn]

this is beautiful

[@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.

[@domm1341]

Brilliant!

[@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

[@oldyoutubevideos7430]

Thanks , it helped a lot!

[@JasonEwton]

This was a really nice video

[@buttsufancypantsu1644]

6:06 when the drugs kick in great video

[@andersonantunes4257]

Very well explained! And very interesting! :D

[@alfiestoppani]

This was the best thing I have ever seen.

[@jkid1134]

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

[@JamesMarek11]

i want to see more of this kind of geometric math, it was very fun.

[@AdityaPrabhu01]

I really like these Geometry videos!

[@Caye2013]

Amazing video! This woman is magical!

[@inspiredtheworld3039]

i really love the way of teaching ... thank you mam

[@mariebcfhs9491]

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

[@MrNarekNers]

Dobar Den Zvezda :D

[@PunmasterSTP]

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

[@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.

[@sacredgeometry]

I love this episode

[@rcookman]

This was useful and interesting.

[@WayneLinnlikestouseGeoGebra]

Numberphile never seize to amaze me.

[@yuhunggoh7628]

Awesome!

[@ArthurStone]

Useful and interesting with great presentation; thank you : -)

[@toxicdesire8811]

yep, best handwriting I've seen on numberphile.

[@shawniscoolerthanyou]

Really going all in on the animation huh? I love it. I wish I could do something like this when teachers ask for proofs.

[@nmmm2000]

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

[@themobiusfunction]

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

[@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.

[@rushabhshah8981]

zvezdelina stankova.... your handwriting is awesome

[@mputuchimezie7966]

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

[@Ensorcle]

Super cool.

[@thydimov9909]

Поздрави от България !

[@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".

[@Reydriel]

Very interesting!