The Many Centres of a Triangle [Peer Review version]

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Where is the centre of a triangle?

It's pretty easy to see where the centre of a square or rectangle is - it's the point where the diagonals cross. Similarly, the middle of a circle is that point your compass pivots around - it's the same distance from the centre to the edge of the circle, whichever direction you take.

But the triangle is different, and it's because triangles come in all shapes and sizes. We have equilateral ones with equal sides, long thin isosceles ones, right-angled ones, and then all manner of wonky ones with different sides and angles1. And it's these wonky ones which have attracted most interest from mathematicians. These describe a triangle in general. Any of the prettier models are just special cases.

In fact, the most interesting thing about the triangle is that it has lots of centres - it just depends how you want to calculate it. This Entry describes a few of them.

The Centroid

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If you draw a line from each point, or vertex, of the triangle to halfway along the opposite side2, then these three lines meet at a point known as the centroid. Mathematicians call these lines medians, and the point at which they all meet is exactly two thirds of the way along each line.

So is this the best definition of the centre? Well, it has two things going for it. If you plot the triangle on a graph, then the centroid is at the average of the x- and y- coordinates of the triangle's vertices. Also, the centroid is the triangle's centre of mass – if you cut out the triangle shape on a piece of cardboard, then you should be able to balance it on the tip of your finger there.

The Incentre

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Two problems with the centroid are that it's generally nearer to one of the vertices and nearer to one of the edges than it is to the others. There are other points, however which are equally distant from these.

The incentre is the point which is the same distance from each edge of the triangle. You find it by drawing the three lines which bisect each of the angles and seeing where they meet.

At the incentre you can draw a circle, the incircle, which fits exactly inside the triangle, just touching each side.

The Circumcentre

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The circumcentre is the point which is the same distance from each of the triangle's vertices. You find it by drawing perpendicular lines from the midpoints of each of the triangle's sides: it's at the point where they all meet.

At this point, you can draw a circle, the circumcircle, which passes through each of the three vertices of the triangle.

Now, this isn't perhaps the best candidate for the triangle's true centre. For obtuse-angled triangles - those with an angle greater than 90 degrees - it's actually located at a point outside the triangle.

The Orthocentre

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If you rotate a triangle so that the side at the bottom is horizontal, then the opposite vertex is its highest point. The vertical line dropping down from this height is called an altitude, and you won't by now be surprised to hear that the three altitudes of a triangle meet at another point, the orthocentre.

Like the circumcentre, the orthocentre lies outside the triangle for obtuse-angled triangles, but it does have some interesting properties. Leonhard Euler discovered that it always lies on a straight line joining the centroid and the circumcentre. Other triangle centres also lie on this Euler line.

The feet of the altitudes have another interesting property. Imagine you have a triangular snooker table. If you chalk a line between two of the feet of the altitudes and hit the ball along it, then it will bounce around three sides and come back to where it started. This is the only path for which that would happen. It's also the shortest path of any kind which visits all three sides of the triangle.

Please don't attempt this with an obtuse-angled triangular snooker table, though; it won't work.

The First Napoleon Point

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Yes, that Napoleon. As well as making his name annexing large parts of continental Europe into the French Empire, Napoleon Bonaparte was an accomplished mathematician. If it was 'Not tonight, Josephine', it could well have been because the great man was developing the famed theorem of geometry which bears his name.

Napoleon's theorem states that if you take any triangle and construct equilateral triangles on the outside of each side, then the centres of these triangles3 form their own equilateral triangle - the outer Napoleon triangle.

If you then join each point of the outer Napoleon triangle with the opposite vertex of the original triangle, then these three lines all meet at the first Napoleon point.

There is also an inner Napoleon triangle and a second Napoleon point. To find them you have to do the same construction, but this time draw the three equilateral triangles on the inside, so that they overlap the original triangle. This is a little bit complicated to draw.

Napoleon's triangles and points are geometrically remarkable, but they don't have much practical significance. Aesthetically, the first Napoleon point looks comfortably central.

The First Fermat Point

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Pierre de Fermat is probably most famous for his Last Theorem' of algebra, an enigma eventually solved in the 1990s by Andrew Wiles. Fermat also has a couple of triangle centres named after him - the first of these is the point which is the shortest total distance from all the vertices.

This centre is actually of practical use. If you have three towns, say, and you need to build a road system joining them, then you will want to save money by using as little road as possible. The first Fermat point is where the three branches of the shortest road system will meet. The roads will be angled at 120 degrees from each other at this point.

It doesn't work for all triangles, though; if the triangle has a vertex with an angle of 120 degrees or more, then the first Fermat point isn't defined. This vertex itself will be the point of intersection for the shortest connecting route.

To construct the first Fermat point, we need to draw the same outer equilateral triangles as we did with the Napoleon Theorem, but this time, join each outer vertex (rather than the centre) to the opposite vertex of the original triangle. The three lines will meet at the first Fermat point.

There is also a second Fermat point, derived in the same way using inner equilateral triangles. It doesn't have the same practical use, and it often lies outside the triangle anyway.

Brocard Points

Imagine three dogs are exercising in a park, when they simultaneously spot each other. The alsatian decides to run towards the beagle, who in turn runs toward the collie, who runs towards the alsatian. When they first notice each other they are at the points of a triangle. As each runs toward its target, that dog in turn is moving towards its own target. In practice the dogs will move in a spiral formation, eventually sniffing bottoms at one of the two Brocard points of the triangle. If instead the alsatian chased the collie, who chased the beagle, who chased the alsatian, then they would meet at the other Brocard point. Mathematicians call their paths pursuit curves.

For Brocard points, we are indebted to the French meteorologist, Henri Brocard (1845-1922). He didn't discover these points by observing dogs; he was actually trying to solve a mathematical problem, to find the point which makes the same angle with two of the triangle's vertices in turn.

As there are two Brocard points, neither can really be treated as a triangle centre, but the Brocard midpoint - halfway between them - could be.

The Nine Point Centre

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One of the most fascinating properties of triangle geometry is that nine geometrically significant points lie on their own circle. The points in question are the three feet of the altitudes, the three midpoints of each of the triangle's sides, and the three points half way between each vertex and the orthocentre. Each is the same distance from a single point, the nine point centre.

This centre happens to lie exactly half way between the circumcentre and the orthocentre. It therefore lies on the Euler line we mentioned earlier.

The nine point circle has interesting relationships with the incircle and circumcircle. It touches the incircle at a single point4, but doesn't cross it. If you draw a line from any point on the circumcircle to the orthocentre, then its midpoint will also lie on the nine point circle.

Despite the name, mathematicians have deduced many other significant points they can construct which lie on the nine point circle.

Other Triangle Centres

{Add image "Euler Line" - Link: https://www.flickr.com/photos/[email protected]/14082220216/}

You might think that we've covered most of the triangle centres you could possibly imagine, but the truth is we've hardly scratched the surface. American mathematician Clark Kimberling has devoted much of his life to collecting them, and his online Encyclopedia of Triangle Centres has faithfully documented the properties of no less than 5,641 of them, at the time of writing.

The term 'centre' isn't particularly accurate - many of these points lie distinctly towards one side or angle of the triangle, like the first Fermat point, or outside it, like the circumcentre and orthocentre can be. These 'centres' are just mathematical functions of the angles and side lengths of the triangle, but ones which have remarkable properties. It must have been a surprise to the first geometer in the ancient world who drew the three medians of a triangle and found they passed through the same point. They then found it also occurred with the three angle bisectors, the three perpendicular line bisectors, and the three altitudes. Each set of lines could have missed each other, but they didn't, and we find it remarkable.

A resurgence of interest in the subject in the 19th Century found many more centres, and computer technology has made it possible to discover more, as well as plot them. They can also help determine those which lie on constructions like the nine point circle and the Euler line.

Further Study

We hope you have found this introduction to this branch of geometry interesting. You'll find plenty of online references, academic papers and books out there if you'd like to study it further, but be warned: triangle centre geometry is a topic that even mathematicians can consider a bit nerdy.

Finally, if you find the fact that a triangle can have many centres somewhat disturbing, then there is a solution: stick to equilateral triangles. They have only one centre: it's two thirds of the way from any vertex to the centre of the opposite side.

1Mathematicians call these scalene triangles.2Commonly called the midpoint of the side.3Equilateral triangles have only one centre, as we'll describe later.4This is named the Feuerbach Point, after the German discoverer of the nine point circle, Karl Feuerbach (1800-1834).

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