A triangle that fits once fits everywhere

Every triangle has two circles attached to it: the circumcircle, through its three corners, and the incircle, touching its three sides. The essays before this one located their centres — the circumcentre on the Euler line, the incentre off it — and one circle touching four found the incircle tangent to the nine-point circle. This one turns the question round. Start with two circles, one inside the other, and ask whether they are the circumcircle and incircle of some triangle.

The natural way to find out is to try. Pick a point on the outer circle, draw a tangent from it to the inner circle, and follow the tangent until it meets the outer circle again. From there draw the other tangent to the inner circle, and continue. After three tangents, either the chain has come back to its starting point — and a triangle fits — or it has not.

The figure below is the case where it does. Three triangles, started from three different points of the outer circle, every one closing after three tangents. The remarkable fact, which Jean-Victor Poncelet proved in 1813 while a prisoner of war in Russia, is that there is no fourth case: if the chain closes from one starting point, it closes from every starting point. Two circles hold either no triangle or infinitely many.

Euler’s distance #

Whether a triangle fits depends only on the two radii and the distance between the centres, and the condition is one Leonhard Euler found in 1765 from the triangle’s side. For every triangle, with circumradius $R$, inradius $r$ and distance $d$ between the circumcentre and the incentre,

$$d^2 = R(R - 2r).$$

The figure below checks this on three hundred triangles of every shape.

Every point lies exactly on the line, and a consequence falls out for free: since $d^2$ cannot be negative, $R \ge 2r$. The circumradius of a triangle is always at least twice its inradius, with equality only when the two centres coincide, which is the equilateral triangle. It is one of the most useful inequalities in triangle geometry, and it is a one-line corollary of a formula for a distance.

Read backwards, Euler’s formula is the condition for the hero’s two circles. Given a circle of radius $R$ and one of radius $r$ inside it, a triangle can have them as circumcircle and incircle only if their centres are $\sqrt{R(R-2r)}$ apart. The hero’s circles, of radii $1$ and $0.38$, are placed at exactly that distance, $0.4899$, and every chain of three tangents closes. The figure also checks that the incentre of each closed triangle is the inner circle’s centre, which it must be.

Off the distance, nothing closes #

Move the inner circle a little, and the porism’s other half appears.

The chain misses its start after three tangents and never comes back. Its corners spread around the whole outer circle, filling it more densely the longer the chain runs. This is not bad luck from one starting point. At this distance no starting point gives a closed triangle, and the chain from every start wanders the same way.

So the two cases are complete and exclusive. At Euler’s distance every start closes after three steps; at any other distance none does. There is no configuration of two circles in which some triangles fit and others do not. That all-or-nothing quality is what the word porism was used for by Greek geometers: a statement that a construction either fails entirely or succeeds with a degree of freedom to spare. Euclid wrote three books of porisms, all lost; what survives of them is Pappus’s description, and the word itself, which later geometers revived for exactly this kind of theorem.

Four sides, and every number of sides #

Nothing in the construction required three. Keep drawing tangents, and ask whether the chain closes after four, or five, or a hundred.

The porism holds for every number of sides. For quadrilaterals, Nicolas Fuss found the condition in 1792:

$$(R^2 - d^2)^2 = 2r^2(R^2 + d^2).$$

Every quadrilateral with an incircle and a circumcircle satisfies it, and conversely, at that distance, every starting point gives a closed quadrilateral. Fuss and later writers found the conditions for five, six, seven and eight sides, each more complicated than the last, and in 1853 Arthur Cayley gave a single criterion for every $n$ at once, as a condition on the coefficients of a power series built from the two circles. Each condition is one equation in $R$, $r$ and $d$, which is the porism again: one equation, and then no further freedom to fail.

Why the distance is what it is #

Euler’s formula has a proof short enough to give, and it uses only the number every chord through a point shares.

Let $I$ be the incentre of a triangle $ABC$ and $O$ its circumcentre. Draw the line from $A$ through $I$; it bisects the angle at $A$, and it meets the circumcircle again at a point $M$, the midpoint of the arc $BC$ opposite $A$. Two facts about $M$ do the work. First, $M$ is exactly as far from $I$ as it is from $B$ and from $C$: the angle $MBI$ and the angle $MIB$ are both half of $A$ plus half of $B$, so the triangle $MBI$ is isosceles. Second, the chord $MB$ subtends half the angle $A$ at the circumference, so by the extended law of sines its length is $2R\sin(A/2)$.

Now $I$ is a point inside the circumcircle, and the chord $AM$ passes through it. The product of the two pieces of any chord through $I$ is the same number, the power of $I$, and for the chord through $O$ that product is $(R - d)(R + d) = R^2 - d^2$. For the chord $AM$ the pieces are $AI$, which is $r/\sin(A/2)$ since $I$ is at distance $r$ from the side $AB$, and $IM = MB = 2R\sin(A/2)$. Their product is $2Rr$ — the sines cancel. So $R^2 - d^2 = 2Rr$, which is Euler’s formula.

The cancellation is worth noticing: nothing about the particular triangle survives. The angle $A$ entered twice and left, which is the algebraic shadow of the porism. A formula connecting $R$, $r$ and $d$ with no trace of the angles is exactly a formula that holds for a whole family of triangles at once.

What every triangle in the family shares #

The triangles between two circles at Euler’s distance form a one-parameter family — one for each starting point on the outer circle — and they have different shapes. The hero’s three triangles are visibly different: one nearly isosceles, one long and thin. What do they have in common, apart from the two circles?

A classical identity answers it. For every triangle,

$$\frac rR = \cos A + \cos B + \cos C - 1,$$

so every triangle in a Poncelet family has the same sum of the cosines of its angles. Their perimeters differ, their areas differ, their angles differ one by one, and the sum of cosines is fixed by the two circles. As the starting point moves round, the angles trade off against each other along a curve in the space of triangle shapes, and that curve is a level set of the cosine sum. At the equilateral end of the range — $r/R = 1/2$ — the level set shrinks to a single shape, the equilateral triangle, and the family is one triangle rotated.

The same holds, with different identities, for the other invariants a family can share. The squared distance between the incentre and the circumcentre is fixed, by construction. The distance from the circumcentre to the orthocentre is not: the Euler line of each triangle has a different length, so the family sweeps its orthocentres around a curve while its circumcentres and incentres stay put.

Two circles that are really five #

The porism for a triangle’s incircle has a sibling for each of its three excircles, the circles touching one side from outside and the extensions of the other two, which one circle touching four drew beside the nine-point circle. For an excircle of radius $r_a$ the relation is $d_a^2 = R(R + 2r_a)$, with the sign changed because the circle lies outside the triangle, and the porism holds with the chain of tangents now crossing the excircle’s side. A triangle’s circumcircle is simultaneously in Poncelet position with four different circles.

Each pairing gives a family of triangles, and the four families through a given circumcircle are related to each other by the same kind of reflection isogonal conjugation used: the excentres and the incentre are the four points fixed by isogonal conjugation, and moving from one to another swaps which of the triangle’s angles is treated as external.

A rotation, seen through a bend #

Why should closure from one start force closure from every start? The chain of tangents defines a map of the outer circle to itself — each corner goes to the next — and the porism says that if some point returns after $n$ steps, every point does. That is exactly what a rotation of a circle does: rotate by one $n$-th of a turn and every point returns after $n$ steps; rotate by any irrational fraction and none ever does.

The tangent map is not a rotation — its steps are long where the inner circle is far from the outer and short where they come close — but it is a rotation in different coordinates. There is a way of measuring angle round the outer circle, stretched in some places and squeezed in others, in which every step of the chain advances by exactly the same amount.

The stretching is visible in where a long chain spends its time. Its corners crowd where the tangent to the inner circle is short — near the side where the circles almost touch — and thin out where the tangent is long, and the crowding follows the curve one over the tangent length exactly. Measure angle so that each small arc counts in proportion to that density, and the chain’s steps become equal: the figure checks eight steps and finds them equal to three decimal places. In that coordinate the tangent map is a rotation by a fixed amount, and a rotation either closes from every start or from none.

That fixed amount is the rotation number of the chain, the average fraction of a turn each step advances.

When the circles are concentric, every step turns through the same angle and the rotation number is that angle divided by a full turn, about $0.403$ for these radii. As the inner circle moves out, the rotation number falls continuously. It passes one third at exactly Euler’s distance — triangles close — and it would reach a quarter at Fuss’s distance, $0.695$, just before the inner circle touches the outer; every fraction in between is reached at a distance of its own. Wherever it is a fraction $p/n$, polygons of $n$ sides close from every start, winding round $p$ times; wherever it is irrational, the chain fills the circle and never returns. The continuous fall of one number explains every porism at once.

Where the rotation comes from #

The existence of the special angle measure is the deep part, and it was Carl Jacobi who found it, in 1828, using elliptic functions.

The pairs of a point on the outer circle and a tangent line from it to the inner circle form a curve of their own — each point has two tangents, and each tangent meets the outer circle twice — and that curve turns out to be an elliptic curve, the same kind of object that governs rows of three points on a cubic. An elliptic curve is, topologically, a torus, and the torus has a natural flat coordinate in which translation is a symmetry. The tangent chain is a composition of two reflections on that curve — swap the point, then swap the tangent — and the composition of two reflections of a torus is a translation. In the flat coordinate it is a fixed shift. Projected back to the outer circle, the flat coordinate is the stretched angle, and the shift is the rotation.

So the porism is a statement about a translation on a torus, and the condition for closure is that the translation has finite order: that $n$ times the shift is a period of the torus. Cayley’s criterion is that condition written out in coefficients. Phillip Griffiths and Joseph Harris gave the modern account of it in 1978, and it is one of the classical theorems whose natural home turned out to be a subject invented long after it.

Poncelet’s prison and projective geometry #

Poncelet was an engineer in Napoleon’s army, captured during the retreat from Moscow in 1812, and he spent 1813 in a prison camp at Saratov on the Volga. Without books he reconstructed the geometry he remembered and pushed it in a new direction, and the notes he brought back became his 1822 treatise on projective properties of figures — the foundation of projective geometry as a subject.

His proof of the porism was projective, not metrical, and it was stated for any two conics, not only circles. A projective map can turn the two circles into two conics of any kind, keeping tangency and incidence, and so closure after $n$ steps is a projective property: the porism for an ellipse inside an ellipse is the same theorem. The circle version is the one with a clean formula, because circles carry the metric information — radii and a distance — that projective geometry forgets.

The same structure appears in the path of a ball on an elliptical table. A ball that never passes between the foci stays tangent to a smaller confocal ellipse, and whether its path closes after $n$ bounces is a Poncelet question between two conics. The periodic billiard paths come in whole families, one through every point of the table’s edge, for exactly the reason the triangles here do.

What the figures cannot certify #

The closing chains are computed in floating point, and closure is checked to a billionth of a unit. A chain at a distance differing from Euler’s by $10^{-12}$ would pass the same check and not close; the figures demonstrate the porism at the distances the formulas give and do not discover those distances. The rotation-number figure estimates the number from three thousand steps at each distance, which pins it to about three decimal places, and the claim that it passes one third exactly at Euler’s distance is Euler’s formula, not the estimate.

The density figure makes an empirical match between a histogram and a formula, and the match is very close; the formula’s correctness is Jacobi’s theorem, and the figure is a check on it rather than a derivation. And none of the pictures shows the torus on which the chain is a translation, because it is a complex curve and lives in four real dimensions.

Still open: polygons in a triangle, and porisms in space #

The porism has been extended in many directions, and some of them are closed while others are not. For Poncelet polygons between two conics the theory is complete: the elliptic curve settles every question about closure, and the conditions for each $n$ are known explicitly.

In three dimensions it is different. Replace the circles by two spheres, or two quadric surfaces, and the chains by polyhedra whose vertices lie on one and whose faces touch the other. Some porism-like statements survive — Emch’s theorem about circles on a sphere, and theorems about quadrics sharing a pencil — but there is no general account of when a polyhedron fitting between two surfaces implies a family of them. The one-dimensional structure that made the plane work, a map of a circle that is secretly a rotation, has no obvious analogue for faces touching a surface, and what the right generalisation of Poncelet’s porism to higher dimensions is remains unsettled.