Nine points on one circle

Take any triangle. Mark the midpoint of each side: three points. Drop a perpendicular from each corner to the opposite side and mark where it lands: three more. Find the point where those three perpendiculars meet, and mark the midpoint between it and each corner: three more.

Nine points, defined by three procedures with nothing obvious in common. They lie on a circle.

There is no reason for that to be true. Nine arbitrary points in the plane do not lie on a circle — three do, four already need a condition, and nine is six conditions past coincidence.

Checking that it is not an artefact of one drawing #

A single well-chosen triangle proves nothing about triangles. So the figure computes the nine points for two hundred and forty further triangles, spread over a range of apex positions, and asserts for each that all nine points are the same distance from the same centre.

That is not a proof either. It is a great deal better than one picture, and its particular value is that it would catch a claim that happened to hold for isosceles triangles or for acute ones and failed elsewhere — which is the commonest way for a geometric coincidence to be a near-coincidence.

The obtuse case is worth drawing because it is where a hasty statement would fail. When a triangle has an angle above a right angle, two of the altitude feet land outside the segments they are dropped onto, and the meeting point of the altitudes lies outside the triangle entirely. Every one of the nine points is still on the circle. The theorem is about lines rather than about segments, and the distinction only becomes visible in the obtuse case.

Nine points, and how unlikely that is #

It is worth putting a number on the surprise before explaining it away.

Three points determine a circle, provided they are not collinear: three unknowns — a centre and a radius — and three equations. A fourth point on the same circle is one extra condition, satisfied by a set of points of measure zero among all possibilities. Nine points on one circle is six conditions beyond the first three.

So the coincidence is not a matter of six numbers happening to be close. It is six exact equalities, and the figure reports them as such: nine distances computed from the coordinates the drawing uses, all agreeing.

Coincidences of this kind are the reason plane geometry survived as a subject long after its results stopped being needed for anything. They are checkable by anyone with a ruler, they are surprising, and — this is the part that matters — they are almost always symptoms of something structural rather than accidents. The nine-point circle is a symptom of a scaling, and finding the scaling is the whole content of the proof.

Why it is true #

The argument is a scaling, and it is short.

Consider the map that shrinks everything towards the meeting point of the altitudes by a factor of a half. It sends each corner to the midpoint between that corner and the meeting point — three of the nine points, immediately.

It also sends the circle through the three corners to a circle of half the radius. So those three points lie on a circle of half the circumradius, centred at the midpoint between the circumcentre and the meeting point of the altitudes.

The other six take one more step each. The midpoint of a side is the image of the corner opposite under the same kind of shrinking towards the centroid, and the altitude feet lie on the circle because each is a point from which a diameter of the small circle subtends a right angle — which is Thales’ theorem applied to the segment between an altitude foot’s two neighbours on the circle.

That last step is the one doing the real work, and it explains why the altitude feet — defined by perpendicularity, which sounds nothing like the other two constructions — end up in the same place. A right angle is a statement about a circle, and the whole subject of the inscribed angle is the dictionary between the two.

The line the centres fall on #

The circumcentre, the centroid and the orthocentre lie on a line, with the centroid twice as far from the orthocentre as from the circumcentre. The figure asserts the collinearity to a billionth of a drawing unit and asserts the ratio to the same tolerance.

The nine-point centre is on that line too, exactly halfway between the circumcentre and the orthocentre, which follows from the shrinking argument above: the small circle’s centre is the image of the big circle’s centre under a halving towards the orthocentre.

The one-to-two ratio is worth a moment on its own. It says the centroid is not merely on the line but at a fixed fraction along it, whatever the triangle — which is the same rigidity that puts the centroid two thirds of the way down every median, and which is what makes the centroid the one classical centre with an elementary formula in the corner coordinates. It is the average of the three corners, which is why the balance point argument works and why nothing similar is available for the orthocentre.

Euler found the line in 1765 and the nine-point circle was noticed in the 1820s, which is a suspiciously long gap given that one is nearly a corollary of the other. The reason is probably that the nine points were assembled from three different traditions — the medial triangle, the orthic triangle and the Euler points — and nobody had put them on the same diagram.

The other triangle hiding inside #

The three midpoints of the sides form a triangle of their own — the medial triangle — and it is a half-sized copy of the original, turned through half a turn about the centroid.

That relationship explains several of the coincidences at once. The medial triangle’s circumcircle is the nine-point circle, which is why the three midpoints are on it. The medial triangle’s own circumcentre is therefore the nine-point centre. And the medial triangle’s orthocentre is the original’s circumcentre, because a half-turn scaling carries altitudes to altitudes.

Chasing that chain is how most of these results are actually proved: find a scaling or a half-turn that carries one configuration to another, and every named point maps to a named point. Nothing is computed and everything follows from one map — which is the same economy that makes Euclid’s shear proof shorter than any calculation of the areas involved.

The orthic triangle — the three altitude feet — is the other one hiding inside, and it is less well behaved: it is not similar to the original, and for an obtuse triangle it does not even sit inside it. Its own connection to the nine-point circle is the Thales argument above rather than a scaling.

What a centre is, and how many there are #

A triangle centre is a point defined from the triangle by a rule that does not depend on how the triangle is labelled or where it sits. The circumcentre, the centroid, the orthocentre and the incentre are the classical four, and each is the meeting point of three lines that had no obligation to meet.

That they meet at all is a separate small miracle each time. Three perpendicular bisectors meet because a point equidistant from A and B and a point equidistant from B and C is equidistant from A and C — which is exactly the argument that makes a Voronoi diagram well defined.

The number of such centres that have been catalogued is now in the tens of thousands, which is either a scandal or a hobby depending on the reader. What separates the four classical ones from the rest is that each has a description that makes it useful elsewhere: nearest-equidistant, balance point, meeting of altitudes, and centre of the inscribed circle. The circumcentre is where a nearest-neighbour partition puts a vertex; the centroid is where a triangle balances; the incentre is the point furthest from all three sides at once. The orthocentre is the odd one out, with no such description — it is defined by a construction rather than by a property, which is why it took the longest to be noticed and why it behaves worst on obtuse triangles.

How the sweep is built, and what it is worth #

The two hundred and forty triangles are not random. They are generated by stepping an apex position through a grid, which makes the figure identical from one drawing to the next — a rule this site applies to anything stochastic, and one that applies here even though nothing is stochastic, because a freshly chosen set of triangles would change what the caption’s number refers to.

Triangles too flat to have a meaningful circumcentre are skipped, and the figure asserts that enough survive for the sweep to mean something. That exclusion is the honest half of the design: a triangle with an area of a millionth has a circumcentre a very long way off, and the nine distances are then large numbers agreeing to a relative precision rather than an absolute one. Including such cases would have meant loosening the tolerance until it stopped testing anything.

What the sweep buys is coverage of shape. The apex wanders across acute, right and obtuse configurations, near-isosceles and thoroughly scalene ones. What it cannot buy is coverage of scale, since all the triangles are drawn at roughly the same size — and the theorem is scale-invariant, so that costs nothing.

Where the coincidences collapse #

The equilateral triangle is the degenerate case, and it collapses everything: circumcentre, centroid, orthocentre and nine-point centre are the same point, the Euler line is undefined because three coincident points determine no line, and the nine points reduce to six because each altitude foot is also a side’s midpoint.

That is the case the figure cannot draw, and the caption says so. It is the standard behaviour of a coincidence theorem — the more symmetric the configuration, the more of the distinct objects fall together, until there is nothing left to be surprised about.

A right-angled triangle collapses less and more instructively. The orthocentre sits at the right-angled corner, so two of the Euler points coincide with two altitude feet, and the nine points reduce to seven. The circle is still there and is still the same circle; two of the labels have merged.

What the theorem is not #

Two readings of the result are wrong in ways worth naming, and both are easy to fall into after looking at the figures.

It is not that any nine interesting points lie on a circle. The incentre’s three touch-points do not lie on the nine-point circle; nor do the three points where the medians meet the sides, which are the midpoints and are already counted. Adding a tenth natural-looking point to the list breaks it immediately. The nine are exactly nine.

And it is not that the circle is somehow the triangle’s own. Two different triangles can share a nine-point circle: reflecting a triangle in the nine-point centre, or taking any of the four triangles formed by three of the four points {A, B, C, orthocentre}, gives a different triangle with the same nine-point circle. So the circle does not determine the triangle, and it does not have a claim to being the circle of anything.

That second fact is the more interesting one, because it says the nine-point circle is an invariant of a configuration of four points rather than of three. Any three of the four have the fourth as their orthocentre, all four triangles share a circle, and the apparent asymmetry of the construction — one triangle, one orthocentre — is an artefact of which three points were named first. The same thing twice motif, again: the object is more symmetric than the description that produced it.

What the picture cannot show #

Every triangle drawn here is scalene and acute or scalene and obtuse. The isosceles case is a shape the sweep passes near and never lands on exactly, and it is the case where two of the nine points come closest to merging without doing so. Nothing here shows the approach to a degenerate configuration, which is where a coincidence theorem is most informative — the objects do not vanish, they crowd together, and the picture stops being able to distinguish them long before the arithmetic does. The same crowding is what makes a dissection proof hard to draw at extreme proportions.

The sweep is two hundred and forty triangles, and the claim is about all of them. The check is strong evidence and it is not a proof, and the proof is the scaling argument, which is four sentences of prose that no drawing here contains.

The tolerance is not zero. The nine distances are asserted equal to within a billionth of a drawing unit on the drawn triangle, and within a millionth across the sweep. Those are the tolerances of floating-point arithmetic rather than of the theorem, which holds exactly. A figure of this kind can never do better than say that a coincidence survives to the precision available.

And no picture shows why the nine points were worth defining. Three of them are natural, three are natural, and the middle three — the midpoints from each corner to the orthocentre — are not something anybody would think to construct without knowing the answer in advance. The theorem is partly a statement about a circle and partly a statement about which nine points to look at, and the second half is history rather than mathematics.

Where the ladder goes next #

The nine-point circle has a further property that is much harder and much stranger: it is tangent to the inscribed circle of the triangle, and to each of the three escribed circles. That is Feuerbach’s theorem, it dates from 1822, and it is the sort of result that makes the subject look like it is hiding something.

There is a computational direction too. Every claim on this page can be turned into polynomial identities in the six coordinates of the triangle, and then verified by algebra rather than by a sweep — which is what a proof by coordinates amounts to, and which is how these results are checked when a sweep is not enough. That method settles a whole family at once and tells nobody why any of it is true, which is the standing trade between a proof that convinces and a proof that explains.

Sideways, the same machinery — a scaling that carries one circle to another — is what generates most triangle coincidences, and the interesting question is which of them are consequences of one another. A great many turn out to be, and the modern treatment replaces the individual theorems with coordinates that make the whole catalogue mechanical.

The other direction leads to what a coincidence means. Nine points on a circle is six conditions holding simultaneously, and the reason it is not miraculous is that the conditions are not independent. Working out which coincidences in a configuration are forced and which are genuinely extra is a question about the equations rather than about the picture, and it is where plane geometry stops being a subject of clever diagrams and becomes a subject of algebra.