One circle touching four

Nine points on one circle ends by naming the result it does not prove: “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.”

The theorem is four statements and each is an equality between a distance and a sum or difference of two radii.

Tangency as an equality #

Two circles touch when they meet at one point, and that is a condition on their centres and radii rather than on a picture. If the centres are $d$ apart and the radii are $r$ and $s$, then the circles touch externally when $d = r + s$ and internally when $d = |r - s|$. Anything else is two intersections or none.

So Feuerbach’s theorem is four equalities. The nine-point circle’s centre is at $|r_9 - r|$ from the incentre, where $r$ is the inradius, and at $r_9 + r_A$ from each escribed centre. The figures compute every one of those distances from the triangle’s coordinates and compare, and the comparison is exact to the arithmetic’s own precision.

That reformulation is what makes the theorem checkable, and it is also what makes it strange. The nine-point circle is built from midpoints, altitude feet and the orthocentre; the inscribed circle is built from angle bisectors. Neither construction mentions the other, and four exact equalities between them is a great deal of coincidence.

The boundary the theorem crosses #

The nine-point circle sets up the reason this result is harder than everything else on its page, and the reason is arithmetic rather than geometric.

Write each centre as a weighted average of the corners. The circumcentre, the centroid, the orthocentre and the nine-point centre all have weights built from the squares of the side lengths, which are polynomials in the coordinates. The incentre’s weights are the side lengths themselves, and a side length is a square root of such a polynomial.

So the first four are rational functions of the six coordinates and the incentre is not. A collinearity among the first four is a polynomial identity, verifiable once and true everywhere; a statement joining the family to the incentre has no such form. Feuerbach’s theorem is a statement across that boundary, which is why no scaling argument reaches it and why its proofs are a page of trigonometry or a well-chosen inversion.

The figures make the boundary visible in a small way. Squaring the tangency condition clears the square roots — $d^2 = (r_9 - r)^2$ is polynomial in the coordinates once both sides are written out, because $r$ appears squared — so the theorem is a polynomial identity after all, at the cost of a squaring that loses which of $r_9 - r$ and $r - r_9$ is meant. That is the usual price of clearing a root, and it is why the algebraic proof is long rather than impossible.

Why four circles and not one #

The inscribed circle touches all three sides from inside. Each escribed circle touches one side from outside and the extensions of the other two, and there are three of them — one opposite each corner. Together the four are the circles tangent to all three lines of the triangle, and there are exactly four such circles.

That is the right way to see why the theorem has four parts rather than one. The nine-point circle is tangent to every circle tangent to the three lines, and tangent to all three lines is a condition with four solutions. So Feuerbach’s theorem is one statement about a family, and the family has four members because a system of three tangency conditions on a circle has four solutions.

The four are distinguished by sign. Writing the inradius and the three exradii as $r$, $r_A$, $r_B$, $r_C$, the identities $r_A + r_B + r_C - r = 4R$ and $1/r = 1/r_A + 1/r_B + 1/r_C$ hold for every triangle — the four radii are not independent, and the relations are what a proof of the theorem for one of them can be transported along.

What the sweep is for #

A configuration of five circles drawn once is suggestive and no more. The figures compute the four tangency conditions on the drawn triangle and then again on a sweep of a couple of hundred further ones, generated by stepping an apex position through a grid.

The value of that is specific: it would catch a claim true of acute triangles and false of obtuse ones, or true of the inscribed circle and false of one of the escribed ones. Those are the commonest ways a tangency result is nearly true, and the obtuse case is exactly where a statement about segments rather than lines breaks.

The figures also check two things the theorem needs and a reader might not think to check. The two circles are not concentric — the tangency is not the degenerate case of one circle inside another sharing a centre — and their radii differ, so $|r_9 - r|$ is not nought. Without those the equality $d = |r_9 - r|$ would be satisfied trivially by two identical circles, and the check would be reporting nothing.

The eight circles it does not touch #

It is worth saying which circles are not in the theorem, because the four that are look like an arbitrary selection until the reason is stated.

The circles tangent to three given lines number four, and that is the whole of the family. The circles tangent to three given circles number eight, which is what eight circles touching three is about, and the count is different because a tangency to a circle has two signs while a tangency to a line has one after the side is fixed.

So Feuerbach’s theorem is about the four-member family and there is no eight-member version of it. The nine-point circle is tangent to the four circles inscribed in the three lines, and asking whether it is tangent to any of the eight circles tangent to three other circles is a different question with no reason to have an answer.

That distinction matters because it says how much of the theorem is a count. Four tangencies is not four separate coincidences to be verified one at a time; it is one statement about a family whose size is decided by a system of equations. The right number of parts is whatever the family has, and the family here has four for a reason that has nothing to do with the nine-point circle.

Where the point of contact is #

The theorem says the circles touch and does not say where, and the point has a name and a construction.

The Feuerbach point is where the nine-point circle touches the inscribed circle. It is the image of the point where the incircle touches… no — the honest statement is that it is characterised as the point where a particular inversion, the one centred at the incentre taking the incircle to itself, carries one circle to the other. That inversion is how the shortest proof of the theorem goes, and it is the same tool a map that trades circles for lines is built on — an inversion turns the tangency into a statement about two lines, where it is elementary.

Three further points arise the same way, one per escribed circle, and all four lie on the nine-point circle. They are catalogued and studied and this essay does not develop them; what is worth taking from their existence is that the theorem is a statement about four points as much as about four tangencies, and the points are where the further structure lives.

Two identities among the radii #

The four radii are not independent, and the relations between them are worth having because they are what a proof for one tangency can be carried along.

Write $r$ for the inradius, $r_A$, $r_B$, $r_C$ for the three exradii, $R$ for the circumradius and $s$ for half the perimeter. Then

$$r_A + r_B + r_C - r = 4R, \qquad \frac{1}{r} = \frac{1}{r_A} + \frac{1}{r_B} + \frac{1}{r_C}.$$

Both are identities in the side lengths, both are a short calculation from $r = \text{area}/s$ and $r_A = \text{area}/(s-a)$, and both say the four circles are tied together.

The first is the one that connects to this page. The nine-point radius is $R/2$, so the first identity reads $r_A + r_B + r_C - r = 8 r_9$ — a relation among the four tangency conditions’ right-hand sides. A proof of one tangency plus these identities does not give the others, since the conditions involve the centres’ positions as well, and that is worth being clear about: the identities constrain the radii and the theorem is about distances.

What the identities do give is a check. The figures compute all four radii from the triangle and could verify both relations; what they verify instead is the four tangencies directly, which is stronger and is the claim the theorem makes.

What the measurement cannot settle #

A tolerance is not zero. The tangency conditions hold to about a part in a thousand million on the drawn triangles, which is the precision of the arithmetic and not of the theorem. A figure of this kind can never do better than report that a coincidence survives to the precision available, and the nine-point circle says the same of its own nine distances.

A sweep is not a proof. A couple of hundred triangles is strong evidence and the theorem is about all of them. What a sweep buys is coverage of shape — acute, obtuse, near-isosceles, thoroughly scalene — and what it cannot buy is the argument.

And the tangency is not explained by anything on this page. The scaling argument that proves the nine points concurrent is four sentences and it does not reach here, because a scaling carries the circumcircle to the nine-point circle and has nothing to say about the incircle. The explanation is either a trigonometric identity or an inversion, and neither is a picture.

What the picture cannot show #

The five circles crowd. On a near-equilateral triangle the nine-point circle and the inscribed circle are close in size and their tangency is a hair’s breadth; on a very flat triangle the escribed circles are enormous and run off any page. The drawn triangles are chosen so that all five fit and none of the tangencies is invisible, and that is a presentational choice with no mathematics in it.

The tangency looks like an intersection. At any drawn resolution a tangency and a very shallow crossing are the same picture, so what the figure establishes is the numbers in its caption rather than the visual impression. That is the standing situation for tangency: it is the one incidence relation a drawing cannot distinguish from its neighbours.

And the equilateral case collapses. There the three escribed circles are congruent and symmetrically placed, the four Feuerbach points are at the midpoints of the sides, and the configuration is more symmetric than the theorem. An earlier essay records the same collapse for the nine points, and it is the standard behaviour of a coincidence theorem.

The escribed circles on their own #

The three escribed circles are the less familiar half of the family and a paragraph on them is worth having, because they are where the obtuse case stops being a special case.

Each escribed circle sits outside the triangle, touching one side and the extensions of the other two. Its centre is the meeting point of one internal angle bisector and two external ones, and the four centres — incentre and three escribed — form a configuration in which each is the orthocentre of the other three. That is the same self-referential shape the nine-point essay notes for the four points $\{A, B, C, H\}$, arriving in the bisector family instead of the altitude one.

Their radii can be large. For a triangle with one very small angle, the escribed circle opposite that corner has a radius growing without bound, so a drawing containing all four is only possible for triangles that are not too flat. That is a drawing constraint and not a mathematical one, and it is why the figures state which triangles they use.

What makes the obtuse case unexceptional here is that the bisector construction never referred to segments. An escribed circle touches the lines of the triangle, and lines do not notice whether a foot lands inside a side. So the tangencies hold for obtuse triangles with no adjustment, where a statement about the altitude feet needs the care the altitude feet are given.

Still open: how much of the catalogue is forced #

Feuerbach’s theorem is one entry in a catalogue of thousands of triangle coincidences, and the interesting question that essay raises is which of them are consequences of one another.

A great many are. The modern treatment writes every centre in weights and turns every claimed incidence into a polynomial identity, which a computer verifies, exactly as a proof nobody can read settles its own question; and the identities turn out to be generated by a much smaller set. What is not settled is a measure of how much of the catalogue is independent — there is no accepted notion of a basis for these identities, and the question of how many genuinely different coincidences a triangle has is not a question anybody knows how to ask precisely.

What is settled is that the rational family and the irrational one behave differently, and Feuerbach’s theorem is the standing example of a statement joining them. Whether there are others of comparable strength — tangencies rather than collinearities, crossing the same boundary — is a question the catalogue answers by enumeration and not by any principle.

The inversion proof, in outline #

The shortest proof is an inversion and it is worth outlining, because it explains where the difficulty went rather than removing it.

Invert in a circle centred at the point where the incircle touches one side. An inversion takes circles and lines to circles and lines, preserves tangency, and can be chosen to send two given circles to two parallel lines. Choose it so that the incircle and one escribed circle both become lines; then the nine-point circle becomes some circle, and the tangencies to be proved become the statement that this circle is squeezed between two parallel lines touching both.

The work is then in identifying what the nine-point circle becomes, and that identification is the page of calculation. The inversion converts four tangency conditions into two, and pays for it by making one of the objects hard to name.

That is the standing trade with inversion, and the map that trades circles for lines is where the trade is set out. It is the right tool when the objects to be simplified are the ones the conclusion is about, and the wrong one when it is the hypothesis that becomes unrecognisable — and Feuerbach’s theorem is close to the boundary, which is why the proof is short and not easy.

The trigonometric proof takes the other route: write everything in terms of the angles, compute both sides of $d^2 = (r_9 - r)^2$, and grind. It is a page, it explains nothing, and it has the advantage of covering all four tangencies at once without any choice of inversion.

What tangency being an equality bought #

A coincidence is easier to believe when it is a number. Nine points on a circle is nine equal distances; four circles tangent to one is four equalities between a distance and a sum of radii. In both cases the reformulation turns a picture into arithmetic, and arithmetic can be checked on two hundred triangles in a moment.

What the reformulation does not do is explain, and the contrast with the nine-point circle is the point. There the nine points are explained by one scaling, and the explanation is shorter than the verification. Here the verification is easy and the explanation is a page, because the result crosses a boundary the easy arguments respect.

That gap between checking and understanding is worth recognising as a feature of the subject rather than a gap in the exposition. A statement can be cheap to verify and expensive to explain, and when it is, the expense is usually a sign that two structures built from different materials are meeting — which here is the difference between a side length and its square.