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三角形有着神奇的高速公路

Triangles have a Magic Highway - Numberphile

我们谈论的是欧拉线,可以说这是对于任何三角形的 “公路” 的一种独特的说法。
We’re talking about the Euler line, which is a special so-to-speak ‘highway’ for any triangle.
你可以对任何三角形画这条公路。
For any triangle you can draw this highway
但是你需要识别一些对于三角形来说特殊的点。
but you need to identify some very special points for the triangle.
你可以在不做标记和不使用圆规的情况下仅仅用直尺画出我们今天所讲的所有东西
You can do everything that we talk about today using only a straight edge with no markings and a compass.
所以这些构造叫做欧几里得结构。
So these constructions are called Euclidean constructions.
让我们开始第一个点。
So let’s start with the first point.
那么你可以取每条边的中点。
Well you can take the midpoints of each of the sides
把他们和对面的顶点连在一起。
and connect them to the opposite vertex.
CD这个部分,有一个特别的名称。
This segment here, CD, has a special name.
它叫三角形的中线。
It’s called the median of the triangle.
好的,所以我们有多少条中线…是三条。
Okay, so how many medians we have… is three.
事实上它在一个三角形中有效但并不保证对所有的三角形都有效。
The fact that this worked for one triangle does not guarantee that it will work for all.
因此,难道我需要在这里画很多的图片并且每一个都试一下吗?
and so, should I draw a lot of pictures here and try on each of them?
或者我应该用一些技巧吗?
Or should I use some technology?
让我们来利用技术。
Let’s use technology.
那么现在让我们来尝试用一个不同的三角。
So let us now experiment having a different triangle.
好的,他们好像交叉了。
Well, they seem to intersect.
唔想象一下,我可以变得疯狂。唔。
Ooh fancy, I can get wild. Ooh.
这样看起来对于任何三角形我们将有这样在一个点中交叉的三个部分。
So it looks like for any triangle we will have those three segments intersecting in a point.
那么这个点就在这一堆的中间。
Well that point turns out to be the center of mass.
这个点还是我们所知道的圆心。
or our point which also is known as the centroid
在东欧它被叫做中线的中间
or in Eastern Europe it is known by the name medicenter
简单来说就是它在三个中线的中间。
which simply says it’s the center of the three medians.
另外一个将会有一个实际的含义
The other point will have some practical implications
假如你已经有了两个村庄然后你想建一条铁路
If you have two villages and you want to build a railroad
那么无论你在铁路上的哪里建一个站点
so that no matter when on this railroad you build a station
两个村庄的居民到火车站都将是一样的距离。
the people from the two villages will walk the same distances to that station.
这就叫AB两个部分的垂直分割线。
It’s called the perpendicular bisector of segment AB.
嗯,无论在铁路的哪里你建立你自己的站点S,
Hmm, no matter where you are on this railroad you build your station S,
两个村庄的居民将会走绝对相等的距离。
People from both villages will walk exactly the same distance.
假设你有三个村庄并且想要建一个学校。
Suppose you have three villages which want to build a school.
人们将把学校建在哪里
Where will they build the school so that kids from all three villages
这样三个村庄的孩子们去学校可以走绝对相等的距离?
walk exactly the same distance to the school?
如果你想A和B走相同的距离
If you want villagers from A and B to walk the same distance
那么你必须在这儿,垂直平分线这里。
then you have to be on there, the perpendicular bisector.
你也可以说,让村庄C和A变得公平一些,
You can also say, about being fair to villages C and A,
这些村庄不得不走在相应的铁路上。
those villagers have to walk on the corresponding railroad.
这个学校应该被建在两个垂直平分线的相交的地方。
The school should be built where the two perpendicular bisectors intersect.
假如我们叫这个点为O,
So if we call this O,
OB等于OA, 并且OA等于OC。
OB = OA, and OA = OC.
(布雷迪)这和中点是不同的对吗?
(Brady) This is different to the med icenter?
是的,它是非常不一样的一个点
Yes, it is a very different one
它有一个名字,叫做外接圆心
and it has a name, it’s called the circumcenter
因为关于他们的一个圆心是O点
because they’re referring to a circle whose center is O
并且这个圆经过这三个顶点
and that circle passes through the three vertices
现在,这个外接圆心是在里面的。
Right now, the circumcenter is inside.
这个时候它经过这个三角形,
The moment it passes the triangle,
他和个三角形看起来是一个直角三角形。
that triangle seems to be a right triangle.
并且它就是一个直角三角形。
And this is true.
如果我们拉出这个外接圆心,那么这个三角形就会变成钝角三角形
Now if we push the circumcenter out, then the triangle becomes obtuse
并且这个特性适用于所有的三角形。
and this is a property which is true for any triangle.
那里对于三个村庄来说也许不是最好的位置
That may not be the most optimal position for the three villages
但是如果你想使这三个村庄变得公平你也许需要把这个点放在外面使它变成一个钝角三角形
but if you want to be fair to all three then you may have to take it out and they form an obtuse triangle
(布雷德)那么那个一个才是三角形真实的中心呢?
(Brady) Which one is the actual center of the triangle?
三角形中没有确切的中心;它取决于你的视角,
There is no actual center; it depends on your viewpoint,
因此它们都应该被称为是三角形的中心。
so both of them deserve to be called centers of the triangle.
所以三角形的面积是底乘以高(我们也叫做顶垂线)除以2
So the area of a triangle is base times height (also called altitude) divided by 2
好的,让我们来查找高的位置
All right, so let’s locate this height
那么我们需要从C点到AB做一条垂直线
So we need to drop a perpendicular from point C to side AB
这是你的直角三角形。
There’s your right angle.
我能从任何位置做它这让我感觉不错
I’m so good I can do it from any position –
我在开玩笑,我将会转动它。
I’m just kidding, I will turn it.
猜一猜会发生什么?
Guess what happens?
这三条高相交在一个点。
Those three altitudes intersect in a point.
这个点,通常用H来表示,它叫做三角形的重心
That point, usually denoted by H, is called the orthocenter of the triangle
在数学中,当你看见一些物体是直角的,一些线,
and in mathematics, when you say that some objects are orthogonal – some lines –
那它们就是垂线.
you really mean they’re perpendicular.
那么我们在这个三角中再一次把这三条垂线,或者高交叉一下,
So we’re intersecting the three perpendiculars, or the three altitudes, in this triangle
他们还是相交在同一个点。
and again, they happen to intersect in the same point.
所以现在,我们要再一次检查是否这不是一个巧合,
So now, we want to again check if this is not a coincidence,
那么让我们来使用一下神奇的软件。
so let us use our magic software.
是的,我们把它在周围移动
Yes, we will move it around
现在看看在我试图让它接近C的时候中心会发生什么
Now watch what will happen with this orthocenter as I try to get it next to C…
这个三角形就会变成直角三角形
That triangle will be right
那么你可以把中心挪到外面,但是它又再一次变成了钝角三角形。
So you can push the orthocenter out, but it will again happen when the triangle is obtuse.
所以这是我们看见的第三个巧合。
So that’s the third coincidence we have seen.
我们有三个不同的中心。
We have three different centers.
我们想知道哪一个点才是最重要的
We’re wondering which one is the most important –
这没有一个好的答案。
there is no good answer to that question.
(布雷德)哪一个是你最喜欢的?
(Brady) Which is your favourite?
重心。
The centroid.
那么现在
So now –
(布雷德)为什们你-等下-为什么你喜欢重心?
(Brady) Why do you – hang on – why do you like the centroid?
啊,因为-哦有一些东西我们无法证明。
Ahh, because – oh there is something we didn’t prove.
这个家伙,AG到GE,的比例是2:1。
This guy, AG… to GE, is like, 2:1.
它是两倍的距离.
It’s twice as long.
然后在另外的中位数上会发生一些相同的事
And the same thing happens on the other medians
他们的比例都是2:1
So they’re all in ratio 2:1
可能在八世纪或者九世纪的保加利亚这个特性是常规几何大纲的一部分
This is a property which was part of the regular geometry program in Bulgaria in maybe 8th or 9th grade
我记得它是用相似的三角形来证明的
and I remember its proof with similar tirangles
我很惊讶你可以用基础的几何工具来证明如此复杂的事实
and I was astounded how you can prove such complicated facts using basic geometric tools
好的,那么现在,如果你有勇气在一个简单的图里把它们三个都画下来-
Okay, so now, if you are brave enough to draw all three in a single picture –
好的,我基本上要闭上我的眼睛然后画这三个点。
Okay, well I’m just going to basically close my eyes and draw three points.
这三个点回在一个特殊的地方吗?
Will those three points lie on a… special place?
(布雷德)啊,对于你来说,它们有可能会的!
(Brady) Ah well, with you, they probably will!
有可能吗?那么让我们试试看!
Probably? Well let’s try!
好的我打算把它们都画在一边。
Okay I’m just going to do it on the side.
“吸气声”
*gasps*
差不多了!好的,我想你们应该画一下。
Almost! Okay, I think you have to do it.
(布雷德)即使你闭着眼睛,你做的数学简直完美!
(Brady) Even with your eyes closed, you do perfect mathematics!
(布雷德)真的?让我们来试试吧。
(Brady) Ready? Here we go.
好的,布雷德尝试去画三个点…(布雷德)闭上-我的眼睛闭上了。
Okay, Brady is attempting to put 3 points… — (Brady) Shut – my eyes are shut.
他基本上画在了一个特殊的位置上
He almost got them in a very special position
但不完全是–(布雷德)不完全是。
But not quite. — (Brady) But not quite.
(布雷德)很少有三个随意的点在一条直线上的
(Brady) Any three random points will rarely be on a line
非常少见,这是一个特殊的情况。
Very rarely, that’s a very special case.
事实是,可能性基本为零。
In fact, of probability zero.
那么这三个中心呢?
How about the three centers?
它们是在一些一般的位置,还是它们与彼此相关联?
Are they in some general position to each other, or are they relatives of each other?
它们是有关系的吗?
They’re related somehow?
所以有中位线的中心,外心,和中心。
So you have the medicenter, the circumcenter, and the orthocenter.
我们有三条中位线,三条垂直平分线,和三条高。
We have the three medians, the three perpendicular bisectors, and the three altitudes.
让我们把它简化一下…
Let’s simplify it…
让我们仅仅看这三个点,让我们看看,发生了什么?
So we are looking only at these three points. Let’s see, what is happening?
那么现在让我们看看是否这三个点都在一条直线上
So now let’s see if these three centers indeed always lie on this line
它们看起来是的。
They look like it.
无论是什么三角形,无论我怎么把这三个点分离,
No matter what the triangle is, no matter how I pull these points apart,
这三个点总是在一条直线上的
the three centers seem always to lie on the line
定理说三角形的三个中心-外心,中位线的中心,和重心-
And the theorem says that the three centers of a triangle – circumcenter, medicenter, and orthocenter –
总是在一条叫做欧拉线的线上。
always lie on the single line called the Euler line.
总是有很多事是真的一旦你知道那里有一条线。
There are lots of things that turn out to be true once you know that there is this line.
三角形中的其他中心也使用中心法则就是说它们都在这条线上
The other centers in the triangle are also defined using certain rules that happen to lie on this line
所以这条线就是三角形的公路
And so this line turns out to be sort of a highway for the triangle
这有一个非常重要的中心叫内切圆心它基本上从来不在这条线上。
There is one very important center called the incenter which almost never lies on this line.
它抵制在那条线上。所以让我们来看看它。
It refuses to lie there. So let’s look at it.
我们尝试在内部画一个和每条边都接触的圆
We will attempt to draw a circle which is inside and it touches each of the sides
需要你做的就是画出角A的角平分线,
What you do is you draw the angle bisector of angle A,
画角B的角平分线,
you draw the angle bisector of angle B,
和角C的角平分线。
and the angle bisector of angle C.
它们相交在一个点。
They intersect in a point.
这是个接触到三角形每条边的中心的圆。
The center of a circle which touches the three sides of the triangle.
所以问题再一次来了,这个内心是在我们的这条路上的吗?
So again the question is, does this incenter lie on our highway?
让我们用电脑检验一下。
Let’s check on the computer.
所以,这条线,它想要在这条线上,但是我们不打算这样做…
So, off the line, it wants to be on the line, but we’re not managing…
现在看将会发生什么当-啊!
Now watch what will happen when – ah!
这就是了,这不仅仅是任何一个三角形。
This is it, this is not just any triangle.
它是一个等腰三角形。
It is an isosceles triangle.
所以这个欧拉线在这种情况下通过顶点C。
And so the Euler line in this case passes through vertex C.
那么它的中位线,高,垂直平分线,再加上额外的角C的角平分线。
And it is the median, altitude, and perpendicular bisector, plus as a bonus the angle bisector of angle C.
所以这就是全部。
So it is all of those.
因此,它也会经过内接圆心,假如它是这个角的角平分线。
And therefore, it will pass through this incenter too, if it is the angle bisector.
但这种情况仅仅发生在等腰三角形中,并且它也不能轻易被证明出来。
But this happens only if the triangle is isosceles, and that’s not easy to prove.
这条路是什么,这个欧拉线,是等边三角形吗?
What is the highway, the Euler line, for an equilateral triangle?
这种情况仅仅发生在所有中心都碰到一起的时候
So this is the only case where all of the centers collide in one
你不能画一天线,因为那仅仅是一个点
And you cannot draw a line; it’s just a point
从现在开始,当你看到三角形中的一个特殊的点的时候,你就可以开始问问题了,
From now on, once you see a special point for a triangle, you can always ask the question,
这个点是不是在欧拉线上?
does this point lie on the Euler line or not?
四分之一大,你就可以看见这一边它的其他兄弟姐妹,这个适用于所有的全等三角形。所以这里会发生什么。
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.

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