Maths prize for 15 July 2016: Treasure Island
Captain Bluebottle has buried his treasure on Triangle Island, and the secret message says only that the treasure is in the exact centre of the island. Where is that?
Vinh Chu, David Trieu, Wei-kong Mao, and Alex On won prizes this fortnight.
David suggested the centre of mass of the triangle (centroid), which is pretty good, since he hasn’t been taught M2.
Vinh suggested the circumcentre (where the perpendicular bisectors of the three sides meet).
Wei-kong suggested the centroid.
Alex had a suggestion all of his own, which was to construct a different triangle from meeting-points of the arcs formed by taking each vertex, in turn, as centre, and each side as radius, and then take the centroid of that second triangle.
I have borrowed this puzzle from Rob Eastaway.
At the event where he presented it, most of us were secondary school teachers, and most of us chose the centroid (see below). I think there are good arguments for that choice.
However, Rob Eastaway also told us that when he has presented the puzzle to primary school teachers, most of them choose the point halfway across the triangle and halfway up.
That choice is definitely wrong, and in fact it’s not a centre at all.
If the triangle is equilateral, all the other definitions of “centre” coincide in one obvious point in the middle of the triangle. But this choice doesn’t.
On this choice, where you see the “centre” depends on which way you look at the triangle – which way up you hold Captain Bluebottle’s map. The definition of the “centre” of a shape should depend only on the shape – not on which way you’re looking at it, any more than on the time of day or the weather or your mood.
Further Maths students will know the centroid as the centre of mass of a uniform lamina (sheet) in the shape of the triangle. You could balance the triangle on a needle placed at exactly that point.
The centroid is also: • One-third up the triangle in each of the three base-to-vertex directions
• The meeting point of the lines drawn from each vertex to the midpoint of the opposite base. (It’s surprising that all three lines meet at the same point, and it has to be proved that they always do; but they do).
When you’re looking to bury treasure, you’re concerned with areas. If the treasure is buried in the centre of the island, then every line drawn through that centre should have equal numbers of possible burying-places rejected to the left of it, as too leftwards, and of possible burying-places rejected to the right of it, as too rightwards. But that is exactly what is true of the centroid: draw any line through it, and the part of the triangle to the left of that line has an area equal to the part to the right of it.
The incentre is: • The centre of the biggest circle that can be drawn inside the triangle
• The meeting point of the bisectors of the three angles of the triangle. (It’s surprising that all three lines meet at the same point, and it has to be proved that they always do; but they do).
• The point which is an equal distance from each side of the triangle (each “coast”, if the triangle is an island).
An equal distance from each coast? Seems quite central. But if the triangle has two long sides and one short side, the incentre is very near the short side, and that doesn’t seem very central.
Further Maths students will have studied centres of mass of wire frames, as well as of laminas (sheets). The Spieker centre is the centre of mass of a wire frame in the shape of a triangle.
It is also the incentre of a smaller triangle drawn by joining the midpoints of the sides of the first triangle.
The Spieker centre is not very far distant from the centroid. Where the centroid is one-third up from each base to the opposite vertex, the Spieker centre is between one-quarter up and one-half up.
The circumcentre is a sort of non-identical twin of the incentre. It is:
• The centre of the smallest circle that can be drawn outside the triangle
• The meeting point of the perpendicular bisectors of the sides. (It’s surprising that all three lines meet at the same point, and it has to be proved that they always do; but they do).
• The point which is an equal distance from each vertex.
The odd thing about the circumcentre is that it can be outside the triangle, in fact a long way outside the triangle. You could say that is a good argument for choosing the circumcentre. Captain Bluebottle has had a good mathematical education. He wants his treasure to be found only by the mathematically-minded, and only the mathematically-minded will think of looking for a centre which is off the coast.
The orthocentre is the meeting point of the altitudes of the triangle, that is, of the perpendicular lines from the vertices to the opposite bases. (It’s surprising that all three lines meet at the same point, and it has to be proved that they always do; but they do).
In some ways it is not quite a “centre”. But it does coincide with the other centres when the triangle is equilateral, and it has a neat relationship with the circumcentre and the centroid.
The orthocentre, centroid, and circumcentre all fall in a straight line called the Euler line, with the centroid between the orthocentre and the circumcentre. The distance between the centroid and the orthocentre is always twice the distance between the centroid and the circumcentre.
The only time all three of these centres fall in the same spot is in the case of an equilateral triangle; and in that case, the incentre is also in the same place.
The orthocentre, like the circumcentre, can be outside the triangle – but it is outside in the opposite direction from the circumcentre (with the centroid, inside the triangle, between them).
There are over 10,000 other definitions of “the centre” of a triangle, catalogued by the US mathematician Clark Kimberling. But the five above are the most common.