Poncelet's Porism

Poncelet's porism states that if an -sided Poncelet transverse constructed for two given conic sections is closed for one point of origin, it is closed for any position of the point of origin. Specifically, given one ellipse inside another, if there exists one circuminscribed (simultaneously inscribed in the outer and circumscribed on the inner) -gon, then any point on the boundary of the outer ellipse is the vertex of some circuminscribed -gon. If the conics are taken as circles (Casey 1888, pp. 124-126), then a polygon which has both an incenter and a circumcenter (and for which the transversals would therefore close) is called a bicentric polygon.

Amazingly, this problem is isomorphic to Gelfand's question (King 1994).

The four panels above show bicentric polygons with 3, 4, 5, and 6 sides, respectively, and only the even-sided panels illustrate the following concurrence result. For the triangle, the blue lines joining vertices to the points of tangency on opposite sides meet at the Gergonne point, while the corresponding lines for the pentagon need not be concurrent.

For a convex even-sided bicentric polygon with nonconcentric incircle and circumcircle, the diagonals joining opposite vertices are concurrent at a limiting point of the two circles (Lachlan 1893, p. 217; Halbeisen and Hungerbühler 2015, pp. 548-549; Garcia et al. 2022). For concentric circles, these diagonals meet at their common center.

Inverting about either of the two limiting points gives two concentric circles. However, the -gonal sides become arcs of circles in the process, so this sort of simple inversion does not provide an automatic proof of the theorem (as happens in Steiner's porism, for example).

Fuss (1792) derived formulas not only for the bicentric quadrilateral, but also the bicentric pentagon, hexagon, heptagon, and octagon, as did Steiner (Fuss 1792; Steiner 1827; Jacobi 1828; Dörrie 1965, p. 192). Chaundy (1923) exhibited porisms for , 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, as well as erroneous expressions for several other values (Kerawala 1947). Richelot derived the expression for . In fact, there is a general analytic expression relating the circumradius , inradius , and offset between the circumcenter and incenter for a bicentric polygon. Given , , and , define

For strictly nested circles, and , so .

Now let

and define the elliptic modulus via

Then the condition for an -gon to be bicentric is

where is a Jacobi elliptic function and is a complete elliptic integral of the first kind (Richelot 1830, Kerawala 1947).

Kerawala (1947) was able to establish many porisms in simple explicit form without resorting to the use of elliptic functions. Cieślak and Szczygielska (2010) demonstrated that porisms may be expressed without the use of elliptic functions for any .

For the two circles illustrated above, with the outer circle taken to be a unit circle, the tangent on the inner circle can be determined by solving

where

is the radius of the inner circle, is the offset of its center, is the given position on the outer circle, and is the angle around the inner circle at which the tangent occurs. Taking the dot product and simplifying gives

When this is solved for , the point at which the extension of this line intersects the outer circle again can be found using the standard equation of a circle-line intersection.

The degrees of the algebraic equations relating , , and for , 4, ..., are 1, 2, 3, 4, 6, 8, 9, 12, 15, 16, 21, 24, 24, 32, 36, ... (OEIS A002348; Kerawala 1947). Let the prime factorization of be written as

where the are distinct odd primes, is a nonnegative integer, and the for are positive integers. For , is given by

In the following expressions, write

following Kerawala (1947), and

following Richelot (1830).

The equation for a bicentric triangle (), i.e., any triangle, may be variously written as

(Richelot 1830), or

(Steiner 1827; F. Gabriel-Marie 1912, pp. 497-501; Kerawala 1947; Altshiller-Court 1952, pp. 85-87; Wells 1992). The latter is sometimes known as the Euler triangle formula.

For a bicentric quadrilateral (), the radii and offset are connected by the equation

(Kerawala 1947), which expands to

(Davis; Durége; Casey 1888, pp. 109-110; F. Gabriel-Marie 1912, pp. 321 and 814-816; Johnson 1929; Dörrie 1965). This can also be written

(Steiner 1827), or

(Richelot 1830).

The relationship for a bicentric pentagon () is

(Steiner 1827) or

(Richelot 1830). A number of alternative forms are given by

and

(Kerawala 1947, pp. 90-95, with the incircle signs).

For ,

(Steiner 1827),

(Richelot 1830),

or

(Kerawala 1947).

For ,

(Jacobi 1828, Kerawala 1947)

For ,

(Kerawala 1947), which can also be written in the form

(Jacobi 1828, Richelot 1830). The equation for a bicentric octagon given by Steiner (1827, p. 289), which is not displayed here, contains at least one typographical error.

For ,

For ,

(Richelot 1830, after correcting the sign before the last left bracket).

For ,

(Richelot 1830, again after correcting the sign before the last left bracket).

For , define , , and , and let and denote the expressions for and evaluated at . Then

(Kerawala 1947, pp. 96-97).

For ,

(Kerawala 1947) or

(Richelot 1830, again after correcting the sign before the last left bracket).

Weill (1878, pp. 278-280) gave a geometric construction with a straightedge and compass that passes from a bicentric polygon with sides to one with sides, allowing the two circles to change.