For any triangle you can draw this highway

but you need to identify some very special points for the triangle.

You can do everything that we talk about today using only a straight edge with no markings and a compass.

So these constructions are called Euclidean constructions.

So let's start with the first point.

Well you can take the midpoints of each of the sides

and connect them to the opposite vertex.

This segment here, CD, has a special name.

It's called the median of the triangle.

Okay, so how many medians we have... is three.

The fact that this worked for one triangle does not guarantee that it will work for all.

and so, should I draw a lot of pictures here and try on each of them?

Or should I use some technology?

Let's use technology.

So let us now experiment having a different triangle.

Well, they seem to intersect.

Ooh fancy, I can get wild. Ooh.

So it looks like for any triangle we will have those three segments intersecting in a point.

Well that point turns out to be the center of mass.

or our point which also is known as the centroid

or in Eastern Europe it is known by the name medicenter

which simply says it's the center of the three medians.

The other point will have some practical implications

If you have two villages and you want to build a railroad

so that no matter when on this railroad you build a station

the people from the two villages will walk the same distances to that station.

It's called the perpendicular bisector of segment AB.

Hmm, no matter where you are on this railroad you build your station S,

People from both villages will walk exactly the same distance.

Suppose you have three villages which want to build a school.

Where will they build the school so that kids from all three villages

walk exactly the same distance to the school?

If you want villagers from A and B to walk the same distance

then you have to be on there, the perpendicular bisector.

You can also say, about being fair to villages C and A,

those villagers have to walk on the corresponding railroad.

The school should be built where the two perpendicular bisectors intersect.

So if we call this O,

OB = OA, and OA = OC.

(Brady) This is different to the medicenter?

Yes, it is a very different one

and it has a name, it's called the circumcenter

because they're referring to a circle whose center is O

and that circle passes through the three vertices

Right now, the circumcenter is inside.

The moment it passes the triangle,

that triangle seems to be a right triangle.

And this is true.

Now if we push the circumcenter out, then the triangle becomes obtuse

and this is a property which is true for any triangle.

That may not be the most optimal position for the three villages

but if you want to be fair to all three then you may have to take it out and they form an obtuse triangle

(Brady) Which one is the actual center of the triangle?

There is no actual center; it depends on your viewpoint,

so both of them deserve to be called centers of the triangle.

So the area of a triangle is base times height (also called altitude) divided by 2

All right, so let's locate this height

So we need to drop a perpendicular from point C to side AB

There's your right angle.

I'm so good I can do it from any position -

I'm just kidding, I will turn it.

Guess what happens?

Those three altitudes intersect in a point.

That point, usually denoted by H, is called the orthocenter of the triangle

and in mathematics, when you say that some objects are orthogonal - some lines -

you really mean they're perpendicular.

So we're intersecting the three perpendiculars, or the three altitudes, in this triangle

and again, they happen to intersect in the same point.

So now, we want to again check if this is not a coincidence,

so let us use our magic software.

Yes, we will move it around

Now watch what will happen with this orthocenter as I try to get it next to C...

That triangle will be right

So you can push the orthocenter out, but it will again happen when the triangle is obtuse.

So that's the third coincidence we have seen.

We have three different centers.

We're wondering which one is the most important -

there is no good answer to that question.

(Brady) Which is your favourite?

The centroid.

So now -

(Brady) Why do you - hang on - why do you like the centroid?

Ahh, because - oh there is something we didn't prove.

This guy, AG... to GE, is like, 2:1.

It's twice as long.

And the same thing happens on the other medians

So they're all in ratio 2:1

This is a property which was part of the regular geometry program in Bulgaria in maybe 8th or 9th grade

and I remember its proof with similar tirangles

and I was astounded how you can prove such complicated facts using basic geometric tools

Okay, so now, if you are brave enough to draw all three in a single picture -

Okay, well I'm just going to basically close my eyes and draw three points.

Will those three points lie on a... special place?

(Brady) Ah well, with you, they probably will!

Probably? Well let's try!

Okay I'm just going to do it on the side.

*gasps*

Almost! Okay, I think you have to do it.

(Brady) Even with your eyes closed, you do perfect mathematics!

(Brady) Ready? Here we go.

Okay, Brady is attempting to put 3 points... -- (Brady) Shut - my eyes are shut.

He almost got them in a very special position

But not quite. -- (Brady) But not quite.

(Brady) Any three random points will rarely be on a line

Very rarely, that's a very special case.

In fact, of probability zero.

How about the three centers?

Are they in some general position to each other, or are they relatives of each other?

They're related somehow?

So you have the medicenter, the circumcenter, and the orthocenter.

We have the three medians, the three perpendicular bisectors, and the three altitudes.

Let's simplify it...

So we are looking only at these three points. Let's see, what is happening?

So now let's see if these three centers indeed always lie on this line

They look like it.

No matter what the triangle is, no matter how I pull these points apart,

the three centers seem always to lie on the line

And the theorem says that the three centers of a triangle - circumcenter, medicenter, and orthocenter -

always lie on the single line called the Euler line.

There are lots of things that turn out to be true once you know that there is this line.

The other centers in the triangle are also defined using certain rules that happen to lie on this line

And so this line turns out to be sort of a highway for the triangle

There is one very important center called the incenter which almost never lies on this line.

It refuses to lie there. So let's look at it.

We will attempt to draw a circle which is inside and it touches each of the sides

What you do is you draw the angle bisector of angle A,

you draw the angle bisector of angle B,

and the angle bisector of angle C.

They intersect in a point.

The center of a circle which touches the three sides of the triangle.

So again the question is, does this incenter lie on our highway?

Let's check on the computer.

So, off the line, it wants to be on the line, but we're not managing...

Now watch what will happen when - ah!

This is it, this is not just any triangle.

It is an isosceles triangle.

And so the Euler line in this case passes through vertex C.

And it is the median, altitude, and perpendicular bisector, plus as a bonus the angle bisector of angle C.

So it is all of those.

And therefore, it will pass through this incenter too, if it is the angle bisector.

But this happens only if the triangle is isosceles, and that's not easy to prove.

What is the highway, the Euler line, for an equilateral triangle?

So this is the only case where all of the centers collide in one

And you cannot draw a line; it's just a point

From now on, once you see a special point for a triangle, you can always ask the question,

does this point lie on the Euler line or not?

One fourth of the big one, and you can see its siblings on the sides, these are all congruent for congruent triangles. So what is happening here.