Name: Triangles have a Magic Highway - Numberphile
Uploaded: 2020-03-27T18:00:46.000Z
Duration: 10 min
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

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

For any triangle you can draw this highway

Adverbs of degree

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.

Well you can take the midpoints of each of the sides

This segment here, CD, has a special name.

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?

So let us now experiment having a different triangle.

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.

(Brady) This is different to the medicenter?

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

that triangle seems to be a right triangle.

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

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

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

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 -

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,

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

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're wondering which one is the most important -

there is no good answer to that question.

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

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

And the same thing happens on the other medians

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!

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

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