Triangles have a Magic Highway
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Triangles have multiple centres, and many of them lie on the so-called Euler Line. More links & stuff in full description below ↓↓↓ Extra footage:
Featuring Zvezdelina Stankova. Editing and animation by Pete McPartlan. Thanks also to Tom Davis of http://geometer.orgMore videos with Zvezda: http://bit.ly/zvezda_videosSupport us on Patreon: http://www.patreon.com/numberphileNUMBERPHILE Website: http://www.numberphile.com/Numberphile on Facebook: http://www.facebook.com/numberphileNumberphile tweets: https://twitter.com/numberphileSubscribe: http://bit.ly/Numberphile_SubNumberphile is supported by the Mathematical Sciences Research Institute (MSRI): http://bit.ly/MSRINumberphileVideos by Brady Haran Brady's videos subreddit: http://www.reddit.com/r/BradyHaran/Brady's latest videos across all channels: http://www.bradyharanblog.com/Sign up for (occasional) emails: http://eepurl.com/YdjL9Numberphile T-Shirts: teespring.com/stores/numberphileOther merchandise: https://store.dftba.com/collections/n...
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We're talking about the Euler line, which is a special so-to-speak "highway" for any triangle.
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For any triangle you can draw this highway
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but you need to identify some very special points for the triangle.
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You can do everything that we talk about today using only a straight edge with no markings and a compass.
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So these constructions are called Euclidean constructions.
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So let's start with the first point.
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Well you can take the midpoints of each of the sides
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and connect them to the opposite vertex.
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This segment here, CD, has a special name.
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It's called the median of the triangle.
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Okay, so how many medians we have... is three.
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The fact that this worked for one triangle does not guarantee that it will work for all.
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and so, should I draw a lot of pictures here and try on each of them?
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Or should I use some technology?
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Let's use technology.
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So let us now experiment having a different triangle.
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Well, they seem to intersect.
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Ooh fancy, I can get wild. Ooh.
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So it looks like for any triangle we will have those three segments intersecting in a point.
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Well that point turns out to be the center of mass.
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or our point which also is known as the centroid
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or in Eastern Europe it is known by the name medicenter
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which simply says it's the center of the three medians.
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The other point will have some practical implications
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If you have two villages and you want to build a railroad
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so that no matter when on this railroad you build a station
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the people from the two villages will walk the same distances to that station.
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It's called the perpendicular bisector of segment AB.
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Hmm, no matter where you are on this railroad you build your station S,
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People from both villages will walk exactly the same distance.
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Suppose you have three villages which want to build a school.
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Where will they build the school so that kids from all three villages
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walk exactly the same distance to the school?
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If you want villagers from A and B to walk the same distance
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then you have to be on there, the perpendicular bisector.
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You can also say, about being fair to villages C and A,
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those villagers have to walk on the corresponding railroad.
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The school should be built where the two perpendicular bisectors intersect.
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So if we call this O,
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OB = OA, and OA = OC.
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(Brady) This is different to the medicenter?
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Yes, it is a very different one
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and it has a name, it's called the circumcenter
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because they're referring to a circle whose center is O
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and that circle passes through the three vertices
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Right now, the circumcenter is inside.
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The moment it passes the triangle,
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that triangle seems to be a right triangle.
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And this is true.
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Now if we push the circumcenter out, then the triangle becomes obtuse
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and this is a property which is true for any triangle.
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That may not be the most optimal position for the three villages
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but if you want to be fair to all three then you may have to take it out and they form an obtuse triangle
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(Brady) Which one is the actual center of the triangle?
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There is no actual center; it depends on your viewpoint,
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so both of them deserve to be called centers of the triangle.
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So the area of a triangle is base times height (also called altitude) divided by 2
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All right, so let's locate this height
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So we need to drop a perpendicular from point C to side AB
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There's your right angle.
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I'm so good I can do it from any position -
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I'm just kidding, I will turn it.
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Guess what happens?
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Those three altitudes intersect in a point.
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That point, usually denoted by H, is called the orthocenter of the triangle
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and in mathematics, when you say that some objects are orthogonal - some lines -
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you really mean they're perpendicular.
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So we're intersecting the three perpendiculars, or the three altitudes, in this triangle
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and again, they happen to intersect in the same point.
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So now, we want to again check if this is not a coincidence,
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so let us use our magic software.
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Yes, we will move it around
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Now watch what will happen with this orthocenter as I try to get it next to C...
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That triangle will be right
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So you can push the orthocenter out, but it will again happen when the triangle is obtuse.
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So that's the third coincidence we have seen.
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We have three different centers.
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We're wondering which one is the most important -
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there is no good answer to that question.
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(Brady) Which is your favourite?
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The centroid.
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So now -
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(Brady) Why do you - hang on - why do you like the centroid?
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Ahh, because - oh there is something we didn't prove.
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This guy, AG... to GE, is like, 2:1.
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It's twice as long.
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And the same thing happens on the other medians
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So they're all in ratio 2:1
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This is a property which was part of the regular geometry program in Bulgaria in maybe 8th or 9th grade
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and I remember its proof with similar tirangles
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and I was astounded how you can prove such complicated facts using basic geometric tools
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Okay, so now, if you are brave enough to draw all three in a single picture -
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Okay, well I'm just going to basically close my eyes and draw three points.
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Will those three points lie on a... special place?
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(Brady) Ah well, with you, they probably will!
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Probably? Well let's try!
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Okay I'm just going to do it on the side.
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gasps
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Almost! Okay, I think you have to do it.
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(Brady) Even with your eyes closed, you do perfect mathematics!
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(Brady) Ready? Here we go.
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Okay, Brady is attempting to put 3 points... -- (Brady) Shut - my eyes are shut.
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He almost got them in a very special position
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But not quite. -- (Brady) But not quite.
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(Brady) Any three random points will rarely be on a line
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Very rarely, that's a very special case.
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In fact, of probability zero.
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How about the three centers?
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Are they in some general position to each other, or are they relatives of each other?
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They're related somehow?
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So you have the medicenter, the circumcenter, and the orthocenter.
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We have the three medians, the three perpendicular bisectors, and the three altitudes.
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Let's simplify it...
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So we are looking only at these three points. Let's see, what is happening?
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So now let's see if these three centers indeed always lie on this line
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They look like it.
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No matter what the triangle is, no matter how I pull these points apart,
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the three centers seem always to lie on the line
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And the theorem says that the three centers of a triangle - circumcenter, medicenter, and orthocenter -
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always lie on the single line called the Euler line.
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There are lots of things that turn out to be true once you know that there is this line.
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The other centers in the triangle are also defined using certain rules that happen to lie on this line
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And so this line turns out to be sort of a highway for the triangle
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There is one very important center called the incenter which almost never lies on this line.
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It refuses to lie there. So let's look at it.
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We will attempt to draw a circle which is inside and it touches each of the sides
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What you do is you draw the angle bisector of angle A,
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you draw the angle bisector of angle B,
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and the angle bisector of angle C.
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They intersect in a point.
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The center of a circle which touches the three sides of the triangle.
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So again the question is, does this incenter lie on our highway?
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Let's check on the computer.
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So, off the line, it wants to be on the line, but we're not managing...
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Now watch what will happen when - ah!
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This is it, this is not just any triangle.
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It is an isosceles triangle.
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And so the Euler line in this case passes through vertex C.
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And it is the median, altitude, and perpendicular bisector, plus as a bonus the angle bisector of angle C.
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So it is all of those.
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And therefore, it will pass through this incenter too, if it is the angle bisector.
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But this happens only if the triangle is isosceles, and that's not easy to prove.
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What is the highway, the Euler line, for an equilateral triangle?
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So this is the only case where all of the centers collide in one
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And you cannot draw a line; it's just a point
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From now on, once you see a special point for a triangle, you can always ask the question,
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does this point lie on the Euler line or not?
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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.