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Current time:0:00Total duration:6:12

Video transcript

so I've drawn 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 there so draw a little bit closer to the midpoint looks like right about there so this 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 which side a B 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 three lines will definitely intersect in a unique point right over there which we call the circumcenter so let me write it so 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 to the midpoint of the opposite side so let me draw so this is a median 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's draw the altitude so I'm going to drop a perpendicular right over here this is one altitude I will draw another altitude so remember it's just going to 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'll try 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 ortho Center ortho ortho Center now the whole reason why I'm doing this and this is frankly kind of a mind-boggling idea it's neat enough that each of these points even exist that if you three altitudes will intersect in one point that the three medians intersect in one point and have all these neat properties and that the three perpendicular bisectors intersect in one point that by itself is neat enough but what's really neat and maybe not completely apparent by the way I drew it because I just drew it freehand is that threes these three points will definitely be on the same line if this was an equilateral triangle they would actually be the same point but 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 two points define a line but three points being on the same line is kind of a you know it's it's it seems like a very unlikely thing if you were to just guess it and I'm going to try to draw it as best as I can if you were to draw this with a straight edge ruler it would come out a lot cleaner but these all three 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's one famous mathematician who tends to get all of the most special and mathematical things named after him because he's the guy who's really explored these things and we call this the Euler line Euler Euler line and I say that Leonhard Euler that doesn't look at what looks like ENTER well I think you get the idea let me write boiler Leonhard 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 for Euler's identity which is e to the I pi is equal to negative one 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 that 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 one seems this very bizarre imaginary number pi is a ratio of the of the diet of the circumference of a circle to its diameter and negative one as well as well negative 1 and so oiler showed that there's a lot of reason to believe that these these four 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 is tells us something kind of crazy and mystical about a reality and if that is not enough for you if you take the midpoint on the euler line between the orthocenter and i'm not going to prove it here i'm just going to give you a little bit of a tidbit of it if you take the midpoint between the orthocenter and the circumcenter so let's take this so I'm going to 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 nine-point circle which intersects this triangle at nine points and we'll see this kind of nine interesting points so let me let me label that as well so it's not even it's cool enough that these three special points runs the euler line but there's actually four 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 nine-point circle and my maybe I'll do another video another video just on that so anyway hopefully you found that kind of interesting