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The magic and mystery of the Euler Line. Created by Sal Khan.
Video transcript
So I've drawn an arbitrary triangle over here that we will assume is not an equilateral triangle And let's draw some of the interesting properties of this triangle So, let's first start with the perpendicular bisectors So, let's say this is the midpoint of this side right over here So let me draw a little bit closer to the midpoint Looks like right about there So thing length is equal to this length right over here And so let me draw a perpendicular bisector So perpendicular bisector would look something like that for that side Now let me do it on this side So, just trying to eyeball it This looks roughly like the midpoint So this length is going to be equal to this length, and then, once again, let me draw a perpendicular bisector So it's going to go at a 90 degree angle I can draw a straighter line than that So that looks pretty good and it is perpendicular And then you can imagine I will do it with this side, with side AB So that looks like the midpoint This length is going to be equal to that length right over there And then if we draw a perpendicular if we draw a perpendicular My best attempt to draw this It looks something like that And we've already learned that these 3 lines will definitely intersect in a unique point right over there which we call the circumcenter So let me write it Let me see This right over there is our circumcenter This is nothing new Circumcenter Now, let's draw the medians So the medians go from each of the vertices to the midpoint of the opposite side So let me draw So this is a median Let me this is a median This is a median And then this is And all the errors in my drawing will start to become apparent now that I'm trying to do multiple things with it But we know all of the medians intersect at the centroid So this right over here this right over here is the centroid Now let's draw the altitudes of this triangle So let me just draw the altitude So I'm gonna draw a perpendicular right over here This is one altitude I will draw another altitude So remember it's gonna go down like this Draw another altitude It looks something like that And then I can draw another altitude from this side, and we know it's going to intersect at this unique point right over here So I tried to draw it as neatly as I can And this is going to be perpendicular although the way I've drawn it doesn't necessarily look that way And we know that this thing right over here is called an orthocenter Ortho orthocenter Now the whole reason why I am doing this and this is frankly kind of a mind-boggling idea It's neat enough that each of these points even exists That if you 3 altitudes will intersect in one point, that the 3 medians intersect in one point and I have all these neat properties, and that the 3 perpendicular bisectors intersect in one point That by itself is neat enough But what's really neat and it's maybe not completely apparent, by the way, I drew it 'coz I just drew it freehand, is that these 3 points will definitely be on the same line If this was an equilateral triangle they would actually be the same point But if for any other triangle, there will be different points and they will be on the same line which is just kind of crazy because obviously 2 points define a line, but 3 points being on the same line is kind of a, you know, it seems like a very unlikely thing if you were just to guess it and I'm gonna try to draw this as best as I can If you're to draw this with as straight edge ruler, it would come out a lot cleaner But with these all 3 points sit on this unique line that, you know, it seems something special or magical about it and because it's something special or magical, there is one famous mathematician who tends to get all of the most special on mathematical things named after him because he's the guy whose really explore these things And we call this the Euler line Euler euler line And I say that Leonard Euler that doesn't look Well it looks like enter Oh, I think you can get the idea when we write Euler Leonard Euler The Euler line The I say that he gets all of the cool and magical, mystical things named after him because he's also responsible for Euler's identity which is e to the i pi is equal to negative 1 And we proved this in the Calculus playlist And if none of this makes any sense, you don't worry We're only in Geometry right now But he gets to have all the This is a magical thing because e comes out of compound interest and growth and decay Exponential growth and decay i is i squared is negative 1 It seems this is very bizarre imaginary number Pi is the ratio of the of the di of the circumference of a circle to its diameter And negative 1 is, well, is, well negative 1 And so Euler showed that there is a lot of reason to believe that these 4 numbers that come from all of these different weird realities of the world are connected in this very tight way And he also showed that these special points are all on the same line which, has tells us something kind of crazy and mystical about our reality And if that is not enough for you, if you were to take the midpoint on the Euler line between the orthocenter, I'm not gonna prove it here I'm just gonna give you a little bit of the tidbit of it If you take the midpoint between the orthocenter and the circumcenter, so let's take this So I'm gonna try to look at it roughly So it looks like it's right about there So this length is going to be equal to this length right over here This point that sits on the Euler line is going to be the center of something called the 9 point circle which intersects this triangle at 9 points And we'll see this kind of 9 interesting points So let me label that as well So it's not even the it's cool enough that these 3 special points are on the Euler line but it's actually 4 special points And actually there's a few more than that that are kind of interesting So that orange point right over there is the center of the 9 point circle And maybe I'll do another video, another video just on that So, anyway, hopefully you found that kind of interesting