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_v...
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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. 🤔

[@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'...

[@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.

[@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.

[@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.

[@JLConawayII]

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

[@Triantalex]

??

[@jordantistetube]

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

[@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.

[@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.

[@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.

[@JDSileo]

I could listen to Professor Stankova lecture all day.

[@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

[@meri7108]

I really love the way Zvezdelina explains things!

[@Henrix1998]

I really like her accent

[@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

[@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.

[@Regular-Sized]

"I can get wild" well that made my day

[@randomusername3388]

1:14 ooh fancy I can get wild ooOoOoh

[@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.

[@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.

[@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

[@TheAAMoy]

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

[@jrgmen]

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

[@jkid1134]

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

[@regulargold7065]

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

[@buttsufancypantsu1644]

6:06 when the drugs kick in great video

[@NerdGlassGamingPA]

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

[@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.

[@spaceminers]

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

[@Latrocinium086]

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

[@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 ;-)

[@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.

[@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.

[@s.d.s.7007]

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

[@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.

[@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.

[@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.

[@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?

[@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 ;)

[@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

[@hovikghazaryan9130]

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

[@lawrencecalablaster568]

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

[@DarkAion]

Витоша )))))

[@pitreason]

Тоже проиграл с этого;)

[@Snioo]

Пародия на Эльбрус? :)

[@Snioo]

О! Это не пародия. Такой компьютер действительно есть.

[@puerto6482]

now now, not very polite

[@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.

[@hasa4628]

I have an app in store that can calculate euler centroid orthocenter and everything about traingle where you simplfy drag the points of traingle whereever you want and get instant result with details steps of how to do it

[@keensauce]

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

[@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

[@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.

[@PunmasterSTP]

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

[@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

[@toxicdesire8811]

yep, best handwriting I've seen on numberphile.

[@migfed]

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

[@DynestiGTI]

My favourite Numberphile video.

[@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.

[@mputuchimezie7966]

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

[@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.

[@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".

[@thomasolson7447]

My favorite video so far.

[@rushabhshah8981]

zvezdelina stankova.... your handwriting is awesome

[@rowdy35967]

Love the animations, well done!

[@РумянаСтанкова-с7ш]

Great presentation and great animation!!

[@FernandoRodriguez-ge2tg]

My favorite video video in a while

[@joeshoesmith]

Me: "I bet it's called the maths line." Video: "...called the Euler line." Me: "Told you so."

[@huskytail]

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

[@WahranRai]

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

[@GAF485]

I studied those centres, or as we called, 'notable points' of a triangle in highschool. Ten years later I still don't know what is notable about the orthocentre! May be it is just the fact that it exist, the centroid, incentre and circumcentre are quite intuitive but the fact that the three altitudes intersects in a common point is not obvious.

[@dco901]

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

[@mariebcfhs9491]

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

[@flux202]

Congratulations on question 6 ma'am👏👏

[@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

[@NeatNit]

I was going to ask "what about the angle bisectors"? And then you went ahead and answered it. Thanks!

[@AlmightyMatthew]

+NeatNit Same. i almost thought they forgot about it

[@MrMarsFromEarth]

The colinear properties of the former 3 centres can be proved by vector operation, regarding which I am quite looking forward to watching a numberphile video.

[@jhoughjr1]

This is one of my favorite numberphile videos

[@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.

[@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!

[@duckofdeathv1595]

Zvezdelina is awesome. Love her videos. Thanks Brady!

[@K31R616]

This lady just blew my mind.

[@LiborTinka]

You can also make a video about Michael Barnsley's "Chaos Game" - a suprising method of constructing a fractal on a paper (Sierpinski's Triangle). This video somehow reminded me of this.

[@aDifferentJT]

There is also the midcircle which is the centre of the circle that passes though the midpoints of the sides and also the intersections of the sides and the altitudes

[@therealelement75]

And the center of that is on the Euler line between the centroid and orthocenter

[@naniapetru]

I learned these things when I was 12 (sixth grade). I'm saying this because it looks like these are advanced definitions that not everyone learns (it seems like this to me because of the way it is explained).

[@overwhelmingsarcasm]

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

[@o3zon]

School build on a railway crossing, seems like a brilliant idea!

[@DanDart]

I was taught only the sides in half method and the 2/3 property at school but was told about its centre of gravity and the angle bisector method, and the dropping perpendicular method in murderous maths

[@jaredj631]

I use this to find the middle of circles often for construction and building things.

[@bakunin888]

The centroid is the actual centre of the triangle! The others are mere impostors! The centroid is by far the most convincing "true centre" if you have to pick one because it is the centre of mass, i.e. the average position of all the points in the triangle, which is also the only physically real centre. This is well demonstrated: in my geometry classes we make cardboard triangles of any shape and find the centroid by drawing lines from the midpoints of each side to the opposite angle, and make a hole. Pin the triangle to the wall through the centroid and it will spin very freely and smoothly. Make a hole at the orthocentre or one of the other centres and notice how the triangle wobbles awkwardly as it spins in comparison.

[@alfiestoppani]

This was the best thing I have ever seen.

[@venkateshbabu5623]

Euler line is convergence of fibbonachi and incenter is a special case of symmetry break.

[@bradzepfan]

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

[@anonymoususer2756]

Medicentre = Bisects line + angle opposite Circumcentre = Perpendicular + bisects line Orthocentre = Angle opposite + perpendicular