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?
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
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.
+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 ?"
+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.
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
So glad this extra footage was here. After watching the previous video at ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-wVH4MS6v23U.html I had to come looking for this.
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" ? :)
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
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.
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.
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.
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.
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... :)
+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.
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!
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".
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.
