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## Geometry (all content)

### Unit 4: Lesson 8

Bringing it all together# Euler line

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-- 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, with side AB. 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-- 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. 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, 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 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 going
to go down like this. Draw another altitude that
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, it doesn't necessarily look that way. And we know that this
thing right over here is called an orthocenter. 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 exists, that three altitudes will
intersect in one point, that the three medians
intersect in one point, and they 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 it's maybe not completely apparent by the
way I drew it, because I just drew it free hand-- is that
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'll 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-- 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 all three points
sit on this unique line that, 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. And I say that Leonhard
Euler-- that looks like ENTER. 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 to 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 squared is negative
1-- seems a very bizarre imaginary number. Pi is a ratio of
the circumference of a circle to its diameter. And negative 1 is,
well, negative 1. And so Euler showed that
there's a lot of reason to believe that 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 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 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 label that as well. So it's cool enough that
these three special points are on the Euler line, but there's
actually four special points. And actually, there's
a few more than that are kind of interesting. So that orange point
right over there is the center of the
nine-point circle. And maybe I'll do another video,
another video just on that. So anyway, hopefully you found
that kind of interesting.