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.