Triangles have a Magic Highway - Numberphile Numberphile English (auto-generated) English

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we are talking about the Euler line which is a special so to speak highway

We're talking about the Euler line, which is a special so-to-speak "highway" for any triangle.

for any triangle for any triangle you can draw this Highway but you need to

For any triangle you can draw this highway

identify some very special points for the triangle you can do everything that we talked about today using only a straightedge with no markings and the

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.

compass so these constructions are called Euclidean constructions so let's

So these constructions are called Euclidean constructions.

start with the first point well you can

So let's start with the first point.

take the midpoints of each of the sides

Well you can take the midpoints of each of the sides

connect them to the opposite vertex this

and connect them to the opposite vertex.

segment here CD has a special name it's

This segment here, CD, has a special name.

called medium of the triangle okay so

It's called the median of the triangle.

how many minions we have is 3 the fact

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

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

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

and so should they draw a lot of pictures here and try on each of them or

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

should I use some technology let's use

Or should I use some technology?

technology to help us know experiment

Let's use technology.

having a different triangle well they

So let us now experiment having a different triangle.

seem to intersect Oh fancy I can get

Well, they seem to intersect. Ooh fancy, I can get wild. Ooh.

wild so it looks like any triangle we will have those three segments

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

intersecting in a point well that point turns out to be the center of mass or our point which also is known the central it or in Eastern

Well that point turns out to be the center of mass. or our point which also is known as the centroid

Europe it is known by the name MIDI

or in Eastern Europe it is known by the name medicenter

Center which simply says it's the centre of the three medians

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

the other point will have some practical

The other point will have some practical implications

implications if you have two villages and you want to build a railroad so that

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

no matter where on this railroad you build a station the people from the two

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

villages will walk the same distance to

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

that station it's called the perpendicular bisector of segment a-b

It's called the perpendicular bisector of segment AB.

hmm no matter where you are on this railroad you build your station as

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

People from both villages will walk exactly the same distance.

have three villages which want to build

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

a school where will they build the school so the kids from all three

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

villages work exactly the same distance

walk exactly the same distance to the school?

to the school if you want villagers from A and B to walk the same distance then

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

you have to be on there perpendicular

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

bisector you can also say how about being fair to villages C and a those

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

villagers have to walk on the

those villagers have to walk on the corresponding railroad.

corresponding railroad the school shall rebuild where the two perpendicular bisectors intersect so if we call this o

The school should be built where the two perpendicular bisectors intersect. So if we call this O,

OB is equal to oay and Oh a is equal to OC this is

OB = OA, and OA = OC.

different to the middie centre yes it is

(Brady) This is different to the medicenter?

a very different one and it has a name

Yes, it is a very different one

it's called the circumcenter because

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

they are referring to a circle whose Center is O and that circle passes through the three vertices right now the

because they're referring to a circle whose center is O and that circle passes through the three vertices

circumcenter is inside the moment it

Right now, the circumcenter is inside.

passes the triangle that triangle seems

The moment it passes the triangle,

to be a right triangle and this is true

that triangle seems to be a right triangle.

now if we push the circumcenter out then the triangle becomes obtuse and this is

And this is true. Now if we push the circumcenter out, then the triangle becomes obtuse

a property which is true or any triangle

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

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

you want to be fair to all three then you may have to take it out if they form

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

an obtuse triangle which one is the actual center of the triangle there is

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

no actual centre it depends on your

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

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

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

derivative is base times Heights also called altitude / -

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

all right so let's locate this height we

All right, so let's locate this height

need to drop a perpendicular from Point C to side a B there's your right angle

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

I'm so good I can do it from any

There's your right angle.

position I'm just kidding overturn it

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

guess what happens

I'm just kidding, I will turn it.

those three altitudes intersect in a

Guess what happens? Those three altitudes intersect in a point.

point that point usually denoted by H is called the orthocenter of the triangle

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

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

you really mean that they are

you really mean they're perpendicular.

perpendicular so we're intersecting the three perpendiculars or the three altitudes in

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

this triangle and again they happen to intersect in the same point so now we

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

want to again check if this is not a

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

coincidence so let us use our magic

so let us use our magic software.

software yes we will move it around now

Yes, we will move it around

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

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

see that triangle will be right so you

That triangle will be right

can push the orthocenter out but it will again happen when the triangle is obtuse

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

So that's the third coincidence we have seen.

seen we have three different centers we

We have three different centers.

are wondering which one is the most

We're wondering which one is the most important -

important there is no good answer to

there is no good answer to that question.

that question which is your favorite the

(Brady) Which is your favourite?

centroid so now what do you think what

The centroid. So now -

do you like the centroid ah because oh there's something we didn't prove this

(Brady) Why do you - hang on - why do you like the centroid? Ahh, because - oh there is something we didn't prove.

guy a G - G is like two to one it's

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

twice as long and the same thing happens on the other medians so they are all in

It's twice as long. And the same thing happens on the other medians

ratio two to one this is a property

So they're all in ratio 2:1

which was part of the regular geometry program in Bulgarian maybe eighth or

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

ninth grade and I remember it's proof with similar triangles and I was

and I remember its proof with similar tirangles

astounded how you can prove such complicated facts using basic geometric

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

tools okay so now if you are brave enough to draw all three in a single

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

picture okay well I'm just going to basically close my eyes and draw three

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

points will those three points lie on a special clay

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

oh well with you they probably we're

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

probably well let's try okay I'm just

Probably? Well let's try!

gonna do it on the side okay huh

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

almost okay I think you have to do it

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

even with your eyes closed give you

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

perfect mathematics ready here we go

(Brady) Ready? Here we go.

okay Brady is at Mt auto shop to put

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

three points he almost got them in a very special position but not quite but

He almost got them in a very special position

not any three random points it's rarely

But not quite. -- (Brady) But not quite. (Brady) Any three random points will rarely be on a line

beyond a line very rarely that's a very

Very rarely, that's a very special case.

special case in fact of probability zero

In fact, of probability zero.

how about the three centers are they in

How about the three centers?

some general position to each other or are they relatives of each other they

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

are related somehow so you have the

They're related somehow?

Medicentre the circumcenter and the orthocenter you

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

have the three medians the three perpendicular bisectors and the three

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

altitudes let's simplify it so we are

Let's simplify it...

looking only at these three points let's see what is happening so now let's see

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

if these three centers indeed always lie

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

on this line they look like it oh no

They look like it.

matter what the triangle is no matter how we pull these points apart the three

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

centers seem always to lie on the line

the three centers seem always to lie on the line

and the theorem says that the three centers of a triangle circumcenter mid center and orthocenter always lie on a single line called the Euler line there

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.

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

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

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

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

very important center called the in-center which almost never lies on

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

this line it refuses to lie there so let's look at it we will attempt to draw

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

a circle which is inside and it touches

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

each of the sides what you do is you draw the angle bisector of angle a you

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

draw the angle bisector of angle B and

you draw the angle bisector of angle B,

the angle bisector of angle C they

and the angle bisector of angle C.

intersect in a point the center of a

They intersect in a point.

circle which touches the three sides of

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

the triangle so again the question is does this incident rely on our highway

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

let's check on the computer off the line

Let's check on the computer.

it wants to be on the line but we are

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

not managing now watch what will happen

Now watch what will happen when - ah!

win ah this is it this is not just any

This is it, this is not just any triangle.

triangle it is an isosceles triangle and

It is an isosceles triangle.

so the Euler line in this case has is

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

through vertex E and it is the median altitude and perpendicular bisector Plus as a bonus the angle bisector of angle 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

So it is all of those.

will pass for this in Center 2 if it is the angle bisector but this happens only

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

if the triangle is isosceles and that's not easy to prove what is the highway

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

the Euler line for an equilateral

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

triangle so this is the only case where all of the centers collide in one and

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

you cannot draw a line it's just a point

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

from now on once you see a special point for triangle you can always ask the

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

question does this point lie on the Euler line or not 1/4 of the big one and you can see it siblings on the sides is all congruent for congruent triangles so

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.

what is happening here

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