Does the euler line give enough information about the triangle it describes to construct a triangle when only given it's euler line? Or is there not enough information inherent in the different points
I came here to ask this same question. Although not of the line itself but the three points specifically. Given the line only (as a complete, extended line) then it wouldn't be so. Obviously if you change the triangle but only enlarge or shrink it and keep the same proportions then the line will stay the same but the three centers will shift position along that line. I think if you have only the line segment terminated by the Ortho- and circum- centers then that should be enough to define two triangles (one with the centroid at 1/3 and another with it at 2/3), and having all three would definitively define only one triangle. But it's hard to tell for sure. I tried to Denise a method of working backwards to get the triangle, but I got lost and couldn't get there. I think it can be done though, I just couldn't figure out how
+puskajussi37 You can't really have a two-gon in euclidean space, because it just degenerates into a line segment... but on the surface of a sphere you can! You could also think of making a monogon on a sphere but I guess that just degenerates into a great circle... which in this geometry is just equivalent to a straight line. Point is... you need curvature to make these things work properly.
13:58 What happens with this 2:1 proof when the triangle "goes" towards an equilateral triangle. Then all the points are on top of each other, and therefore the ratio must be different when that is the case, and just before that is the case as well?? Anyone??
When you have an equilateral triangle the 2:1 relationship is technically still valid. Think of it like this: the distance from the Orthocenter to the Medicenter is always 2 times longer than the distance of the Circumcenter to the Medicenter. Simpler put "CM = 2*OM" or "CM/2 = OM" whichever you prefer. in an equilateral triangle the distances are obviously zero, so the equation goes "0 = 2*0" -> "0 = 0" or "0/2 = 0" -> "0 = 0" which are both true statements, thus the rule is not broken. Now if you were to shift any of the three points of an equilateral triangle in any direction by any Infinitesimal length "X" (a number so small that it is basically 1/∞), then the Orthocenter and Medicenter would move apart the exact same Infinitesimal length "X" and the Circumcenter and Medicenter would move apart exactly half that Infinitesimal length or "X / 2".
A language question about a word that I hear frequently in math/physics context, but cant't get the meaning of. It sounds like "guy" or "guide"; in this video it's used e.g. at 6:04. Help, anyone? spelling? proper uses? thx!
For non-Euclidian (positively or negatively curved space) triangles, is the Euler line in some sense straight or are they curved, or do they even have an Euler line at all?
+Omnis Imperator I'm not 100% sure, but I would guess no. Since the distances are always in ratio 1:2, they only information the centres give you is an overall scale. That one piece of information isn't enough to uniquely define a triangle, you'd need three numbers to do that.
+Omnis Imperator easy No, the easiest case is the 3 centers lying on the same point - insta showstopper. When thinking about the other cases, it seems to me that you can figure out the dimensions if you know the centers plus either 1 angle or 1 side length (and which angle/side is meant). Can't prove it though
@@niksxr You can rotate, scale, and translate *any* non equilateral triangle so that all three centers coincide with another non-equilateral triangle. Because this gives you almost no information about the triangle, you almost definitely need two pieces of information.
@@niksxr No, you need a minimum of two. If you were given just one angle, for example, the other two angles can take on a whole range of values. But, for any of those potential triangles, you know you can rotate and scale them so their centers lie on the same three points as what you were given. That means you haven't distinguished between all those possibilities.
Are there 3 dimensional shape's with similar properties? I ask because the animations make the magic highway look like it creates a Z axis perpendicular to an equilateral triangle which is rotating in 3 dimensional space. It could be cool to see a similar effect on an already 3D shape :3
9 point circle, center of the circle running through the midpoints of each line, where the altitudes intersect, and the midpoint between the vertices and the orthocenter
You know that you are old when watching this brings up memories which seem to you like being from a time when dinosaurs roamed the earth. I feel so old right now.
Just how they produce frequency. Some kind of following Euler lines. So maybe spheres produce frequency on those points for something like earth. That's why tides are in pressure gradients. Mostly stress lines for quakes. When they move.
Well, all triangles whose vertices don't lie on a circle also don't have an orthocentre. It just so happens that there are no such triangles but still.
Hmmm... I don't know if I (or someone else) should be concerned by the fact that this is the fifth time (or sixth... honestly I've lost count) I watch this video. Every time I restart my browser it reloads the "previous session" and I usually start by just closing the old pages that I have already "read/watched" but every time I come to this one I watch it again... And leave it for the next "session"... Maybe I'm in need of an intervention, am I the first "documented case" of "numberphilea" ? :)
So, can we take a triangular prism with a triangular base and stick a sphere in it and around it and poke some lines through it with similar results? I kinda want to try.
+Mateusz Szafrański For some reason I imagine "knowlage" being pronounced French, while swirling some red wine. "Quelle facilité à délivrer du no-laaaaaage, n'est-ce pas ?"
What is the practical purpose of the orthocentre though? The centroid is the geometric centre, the circumcentre is the centre of the circumcircle, and the incentre is the centre of the incircle. But where does the orthocentre "come from", other than just being the intersection of the 3 altitudes?
orthocenter is very much related to circumcenter. First thing we know are from this video, orthocenter is circumcenter reflected twice about the medicenter. Second: if you reflect each altitude about the corresponding angle bisector, the three reflections meet at the circumcenter, that's why orthocenter & circumcenter are called "Isogonal Conjugates". Third: when you reflect Orthocenter about the three sides of the triangle, the three reflections all lie on the circumcircle. So in a way you are right, orthocenter is just some derivative of circumcenter.
What I would now like to know is this: do these centres (or other centres) and the Euler line exist for a triangle on a non-euclidean surface, and if so, how does the curvature (whether positive or negative) of the surface affect the centres and Euler line?
+Jane Stacey I'm just guessing here from the top of my head, that they (the centres) will exist and indeed there will be an Euler line. But it will of course follow the curvature, the same way as the medians and all other lines will do. In fact curvature will have nothing to do with it. Take a piece of fabric, draw the shapes and lines and just play with the cloth... :)
The Golden triangle seems to have a lot of unique properties involving the Golden ratio itself.. I'm wondering if possibly the Euler line's points lies in a Golden ratio fashion as well...
Apparently the tryangle allows spirits to come through from another demention! Jayz and so many in the music industry use this symbol. Satan can use this as an energy feild but so does The LORD! The devil is always trying to userp GOD'S power.
Marvelous demonstration.
The Euler line looks like a three-dimensional normal to the triangle face.
+DrSegatron I can see that, too. It looks like a triangle pierced by the euler line. It becomes really clear when the shape is changing.
Such perception delusion is not limited to Euler's line.
No worth in pointing it out every time.
it is,and the proof can be much simple in 3 dimensions
In fact all of the triangles look like a shadow cast onto a piece of paper by an equilateral triangle being rotated about in 3D space!
I love this lady! !
+Hari Krishnan Me too, man, and I don't even know why...
'cause she's a star!
same! cute enthused energy
Thank you for these videos. I'm thinking seriously to support you on Patreon!
Professor Stankova is truly a gifted teacher.
This was just beautiful
When she said "And that unique point... is the Centroid.", I had a very steep inclination towards saying "I love you"
Could you do this with 3-D objects like, say, a triangular-based pyramid?
Lots of fun to load up CARMetal and play with this.
Can a similar line be created with a pyramid?
Does the euler line give enough information about the triangle it describes to construct a triangle when only given it's euler line? Or is there not enough information inherent in the different points
I came here to ask this same question. Although not of the line itself but the three points specifically. Given the line only (as a complete, extended line) then it wouldn't be so. Obviously if you change the triangle but only enlarge or shrink it and keep the same proportions then the line will stay the same but the three centers will shift position along that line.
I think if you have only the line segment terminated by the Ortho- and circum- centers then that should be enough to define two triangles (one with the centroid at 1/3 and another with it at 2/3), and having all three would definitively define only one triangle. But it's hard to tell for sure. I tried to Denise a method of working backwards to get the triangle, but I got lost and couldn't get there.
I think it can be done though, I just couldn't figure out how
11:10 Triforce
:l
I think she meant "triangle", not "circle" in 1:40
What about a Euler plane surface for pyramids? And is there a Euler pyramid for 4 dimensional "pyramids"?
Does an euler line appears also in Non-Euclidian spaces??
Could a closed line be considered a two angled polygon? Would monogon be a point, a line with one end or an anfinite line with one bend in it?
+puskajussi37 You can't really have a two-gon in euclidean space, because it just degenerates into a line segment... but on the surface of a sphere you can! You could also think of making a monogon on a sphere but I guess that just degenerates into a great circle... which in this geometry is just equivalent to a straight line. Point is... you need curvature to make these things work properly.
Free Dell advert
Wow
Saturn turn is the e. The Euler numbers mess.
13:58 What happens with this 2:1 proof when the triangle "goes" towards an equilateral triangle. Then all the points are on top of each other, and therefore the ratio must be different when that is the case, and just before that is the case as well?? Anyone??
I think the equilateral triangle is only the case, but for the triangles before that it's still 2:1
When you have an equilateral triangle the 2:1 relationship is technically still valid. Think of it like this: the distance from the Orthocenter to the Medicenter is always 2 times longer than the distance of the Circumcenter to the Medicenter. Simpler put "CM = 2*OM" or "CM/2 = OM" whichever you prefer.
in an equilateral triangle the distances are obviously zero, so the equation goes "0 = 2*0" -> "0 = 0" or "0/2 = 0" -> "0 = 0" which are both true statements, thus the rule is not broken.
Now if you were to shift any of the three points of an equilateral triangle in any direction by any Infinitesimal length "X" (a number so small that it is basically 1/∞), then the Orthocenter and Medicenter would move apart the exact same Infinitesimal length "X" and the Circumcenter and Medicenter would move apart exactly half that Infinitesimal length or "X / 2".
Kiba Nemial Wow, that does work. Thanks
A language question about a word that I hear frequently in math/physics context, but cant't get the meaning of.
It sounds like "guy" or "guide"; in this video it's used e.g. at 6:04.
Help, anyone? spelling? proper uses?
thx!
+ozdergecko It's "this guy". It's just a colloquial demonstrative pronoun for a thing.
en.wiktionary.org/wiki/guy#Etymology_2
+ozdergecko seawas! "This guy" = "Der do"
Schindlabua jössas, der. jo, eh! pfiat enk, sogt da peda
eyy it's my professor!!
+Butt McFarts
You lucky sob.
congrats man. make the most of it.
I kinda have a crush on this lady...
Yes
Me too
The triangle is the best polygon. I learnt this from vihart
triangle. triangle triangle triangle triangle triangle
+lai yong hui it's even an instrument
+lai yong hui i know it is my favorite since my school years, though i never had a reason :)
+Giorgos Chatziioannou (LezantasGR34T) Same. But when I showed these 'triangles' to friend, they don't care about it
+lai yong hui Please, everyone knows the octagon is the most holy of shapes.
that was like watching a mystery show. brilliant.
I love Zvezda.
For non-Euclidian (positively or negatively curved space) triangles, is the Euler line in some sense straight or are they curved, or do they even have an Euler line at all?
+TimeAndChance If I remember from my college days properly, I don't believe there is a Euler line at all in non-Euclidian geometry.
+TimeAndChance I'm afraid that in non-Euclidean planes the three centers might not even exist (i.e. the lines defining them might not coincide).
the center of mass exists for sure,same place with constant curvature,varying place with locally varying curvature
How does every person on this channel have a wonderfully pleasant voice and accent !?
They are chosen specially
Can you use the 3 points on the line to work out the dimensions of the triangle?
+Omnis Imperator I'm not 100% sure, but I would guess no. Since the distances are always in ratio 1:2, they only information the centres give you is an overall scale. That one piece of information isn't enough to uniquely define a triangle, you'd need three numbers to do that.
+Omnis Imperator easy No, the easiest case is the 3 centers lying on the same point - insta showstopper. When thinking about the other cases, it seems to me that you can figure out the dimensions if you know the centers plus either 1 angle or 1 side length (and which angle/side is meant). Can't prove it though
@@niksxr You can rotate, scale, and translate *any* non equilateral triangle so that all three centers coincide with another non-equilateral triangle. Because this gives you almost no information about the triangle, you almost definitely need two pieces of information.
@@MushookieMan thanks, yes. But I think you just need one more defined piece of the triangle to define it all
@@niksxr No, you need a minimum of two. If you were given just one angle, for example, the other two angles can take on a whole range of values. But, for any of those potential triangles, you know you can rotate and scale them so their centers lie on the same three points as what you were given. That means you haven't distinguished between all those possibilities.
Math just never gets old. Thank you, always, for these learning tools!
Are there 3 dimensional shape's with similar properties? I ask because the animations make the magic highway look like it creates a Z axis perpendicular to an equilateral triangle which is rotating in 3 dimensional space. It could be cool to see a similar effect on an already 3D shape :3
I've never been so into geometry as after watching this.
which program was used to do that triangle?
+Icaro Vasconcelos I don't know which particular program this is. Though Geogebra can do the same and more.
MS Paint
Illuminati confirmed?
+RanEncounter 3 centres on the Euler line. HALF LIFE 3 CONFIRMED!
You aren't gonna last long in a geometry class with that habit.
Is there a special name for the point halfway between the centroid and the orthocenter on the Euler line? (13:37 unnamed point between G and H)
i think its name is bill
I wanted to know this too. It ought to have a name but they don't mention it. 🤔
Look it up, you may be able to give it a cool name
9 point circle, center of the circle running through the midpoints of each line, where the altitudes intersect, and the midpoint between the vertices and the orthocenter
What a great mathematician!
+Caye2013
Leonhard Euler or Zvezdelina Stankova?
+Penny Lane
Both hahaha
You know that you are old when watching this brings up memories which seem to you like being from a time when dinosaurs roamed the earth. I feel so old right now.
15 more minutes with this beautiful lady
Just how they produce frequency. Some kind of following Euler lines. So maybe spheres produce frequency on those points for something like earth. That's why tides are in pressure gradients. Mostly stress lines for quakes. When they move.
I want to marry her, seriously
Well, all triangles whose vertices don't lie on a circle also don't have an orthocentre. It just so happens that there are no such triangles but still.
funny...
AscendingApsolut No, just a set-theoretically true statement ...
It doesn't make it less funny.(to me)
AscendingApsolut Well ok, then I'm glad you enjoyed yourself :)
*;)*
What's the Euler line for an equaliteral triangle? Is it just a point?
Well the SAS similarity was bogus similarity is shown by angle not side length and it's much easier too
Beautiful proofs. This professor explains the proofs very well. I remember a harder proof that the Euler line exists--so I learned something new.
those freehand straight lines, tho
OMG, I'm in love!
more camera focus on the paper, and not on the person, this gives me a headache. this is a main problem with num philes.
A wonderful and useful presentation!
You are the most brilliant but more importantly the sweetest mathematician I have ever seen and all that makes you a MIRACLE!!
"you realize some coincides are theorems, and then you try to prove them"
Mind = Blown🎇🎆🎆🎇🎆🎇🎆
Nice proof :)
I so love the pro. Does she write some math books?
Hmmm... I don't know if I (or someone else) should be concerned by the fact that this is the fifth time (or sixth... honestly I've lost count) I watch this video. Every time I restart my browser it reloads the "previous session" and I usually start by just closing the old pages that I have already "read/watched" but every time I come to this one I watch it again... And leave it for the next "session"... Maybe I'm in need of an intervention, am I the first "documented case" of "numberphilea" ?
:)
Zvezda has such nice printing.
Numbers cannot lie.
So glad this extra footage was here. After watching the previous video at ua-cam.com/video/wVH4MS6v23U/v-deo.html I had to come looking for this.
The triangle can be described and represented as three points on a line. SO you can draw a line, put points on it and have a triangle.
When looking at the animations, I realized that it looks like the line is on a perpendicular with the triangle itself, like a 3D object.
She has a nice handwriting.
Excellent presentation. Visit us in South Africa: you will change the image of math.
At 2:34 there's a sound of a piano key in the back ground, it's also in both the hook number videos, what's going on?!
So, can we take a triangular prism with a triangular base and stick a sphere in it and around it and poke some lines through it with similar results? I kinda want to try.
No you need to use a tetrahedron. A shape made.up of 4 triangles. It's also known as a triangular pyramid.
don't do it,people who did all died
11:25 triforce! °o°
What a great easiness of delivering knowlage!
+Mateusz Szafrański For some reason I imagine "knowlage" being pronounced French, while swirling some red wine. "Quelle facilité à délivrer du no-laaaaaage, n'est-ce pas ?"
Beautiful mind
What is the relation between the largest spheres that could fit within pyramids with the triangle as the base, and the centers as the ceilings?
Sorry Pete, but you missed a 'z' on young Zvezda's geometry notes! :p
marry me professor.
How do we know that the orthocenter of the little triangle is the circumcenter of the big triangle?
i understabnd ED=1/2 AB
i dont understand GD=1/2 AG fast
EG BG джи джи биджи
Isnt a poont just a representation of a line perpendicular to the plane
should link a pdf explaining the math behind these videos
I knew Zelda emblem was going to show up here
Why is the camera continuously moving?
What is the practical purpose of the orthocentre though?
The centroid is the geometric centre, the circumcentre is the centre of the circumcircle, and the incentre is the centre of the incircle.
But where does the orthocentre "come from", other than just being the intersection of the 3 altitudes?
orthocenter is very much related to circumcenter. First thing we know are from this video, orthocenter is circumcenter reflected twice about the medicenter. Second: if you reflect each altitude about the corresponding angle bisector, the three reflections meet at the circumcenter, that's why orthocenter & circumcenter are called "Isogonal Conjugates". Third: when you reflect Orthocenter about the three sides of the triangle, the three reflections all lie on the circumcircle. So in a way you are right, orthocenter is just some derivative of circumcenter.
There are more than 200 triangle centers
relike868p Really?
en.wikipedia.org/wiki/Encyclopedia_of_Triangle_Centers
Does the tetrahedron have some type of euler surface?
What I would now like to know is this: do these centres (or other centres) and the Euler line exist for a triangle on a non-euclidean surface, and if so, how does the curvature (whether positive or negative) of the surface affect the centres and Euler line?
+Jane Stacey I'm just guessing here from the top of my head, that they (the centres) will exist and indeed there will be an Euler line. But it will of course follow the curvature, the same way as the medians and all other lines will do. In fact curvature will have nothing to do with it. Take a piece of fabric, draw the shapes and lines and just play with the cloth... :)
¿what about their ratio?
see till the end prior to comment
can we build a smaller congruent triangle with the 3 centers of the bigger triangle?
nice question. suppose not really a smaller one, but at any case one mirrored about / over the euler line.
she's such a brilliant teacher
I would marry a woman Iike this. I don’t care if she’s older or not. Intelligent, passionate, educated women > all others.
The Golden triangle seems to have a lot of unique properties involving the Golden ratio itself.. I'm wondering if possibly the Euler line's points lies in a Golden ratio fashion as well...
i love the lady :D
Apparently the tryangle allows spirits to come through from another demention! Jayz and so many in the music industry use this symbol. Satan can use this as an energy feild but so does The LORD! The devil is always trying to userp GOD'S power.