Six points on a conic, and the line they share

The classification of the general quadratic ended with a count. A conic has six coefficients, and scaling all of them gives the same curve, so it is fixed by five numbers — and each point it must pass through imposes one condition. Five points determine a conic. So a sixth point either lies on the curve through the other five or it does not, and there ought to be a way to say which.

The count says nothing about what that way is. Blaise Pascal, at sixteen, found an answer that uses no coefficients at all.

Six points lie on one conic exactly when the three pairs of opposite sides of the hexagon they make meet in three points on a line. The figure computes each meeting point as a cross product of lines in homogeneous coordinates, measures the three for collinearity, and then moves each corner in turn off the curve and requires the largest of those moves to break the alignment decisively — so the check is one the construction could fail.

A statement with no measurements in it #

Every description of a conic in the earlier essays has measured something. Cutting a cone fixes an angle; the focal definitions fix distances; the family that crosses at right angles is about an angle again; the discriminant is arithmetic on coefficients. Pascal’s theorem measures nothing. It is about which points lie on which lines — incidence, and nothing else.

That has a consequence that makes the theorem far stronger than it looks. A central projection — casting a shadow of a flat figure from a point onto another plane — can change every length and every angle, but it sends points on a line to points on a line and a conic to a conic. So if the theorem is true for one conic it is true for every conic that is a shadow of it, and every conic is a shadow of a circle: that is what the cone was. A theorem about incidence proved for the circle is proved for the ellipse, the parabola and the hyperbola at once.

The circle version has a short proof once one more freedom of projection is used. Project so that the line through two of the meeting points is sent off to infinity. Then two pairs of opposite sides have become parallel, and the claim is that the third pair is parallel too. On a circle that is a statement about arcs: parallel chords cut off equal arcs between them, and two such equalities around the hexagon force the third. Undo the projection and the three meeting points are back on a finite line.

The determinant the hexagon replaces #

There is a purely algebraic test for six points, and setting it beside Pascal’s shows what the theorem buys.

Each point $(x, y)$ gives a row $(x^2, xy, y^2, x, y, 1)$. Six points lie on one conic exactly when some set of six coefficients, not all nought, makes all six rows vanish — which is to say exactly when the six rows are dependent, and the six-by-six determinant built from them is nought. The determinant measures how much room a set of rows leaves, and here it is nought when the sixth condition adds no room at all. The straightedge figure below uses the five-row version of the same idea to find its conic: the six coefficients are the signed five-by-five minors of the five rows.

That test is complete and correct, and it is opaque. Written out it has 720 terms, each of degree eight in the coordinates, and nothing about its vanishing says what the six points look like. Pascal’s theorem is the same condition — the determinant is nought exactly when the three meeting points are collinear — made into a picture that can be drawn with a ruler. One polynomial of degree eight in twelve numbers is a statement about three points on a line.

The two tests also differ in what they survive. Under a change of coordinates the determinant changes, and only the fact that it is nought is preserved, which has to be checked. Pascal’s condition is preserved by construction, because every map that sends lines to lines carries three collinear points to three collinear points. An ellipse is a stretched circle and a parabola is a circle seen in perspective, and in both cases the hexagon’s sides go to the new hexagon’s sides and the theorem comes along. The algebra has to be shown to be invariant; the geometry cannot help being so.

Every hexagon, even a crossed one #

The theorem does not ask the hexagon to be convex, or even to be a sensible polygon. Six points can be visited in any order, and each order gives a hexagon with its own three pairs of opposite sides.

Six labelled points can be joined into a closed hexagon in sixty different ways, counting a hexagon and its reversal as one. Each has a Pascal line, so six points on a conic carry sixty lines, every one of them forced. That family has more structure than the theorem itself: Jakob Steiner showed in 1828 that the sixty lines pass three at a time through twenty points, and Thomas Kirkman found sixty more points where they meet in threes.

The crossed hexagon also shows why the control moves every corner rather than one. Moving the sixth corner of this hexagon off the curve shifts the three meeting points by almost nothing, because of where that corner sits relative to the others; moving the fourth breaks the alignment clearly. The figure reports the largest effect, and it requires that some single corner off the curve break the line decisively.

Five points and a straightedge #

Read the theorem backwards and it becomes a construction. Given five points, a sixth point on the same conic is exactly a point that makes the Pascal condition hold — and the condition can be met with nothing but a straightedge.

The construction is Pascal’s theorem with the sixth point unknown. Call it $X$, on a chosen line through point one. The hexagon is 1, 2, 3, 4, 5, $X$. Its first pair of opposite sides, 12 and 45, meets at a point A that does not involve $X$ at all. Its third pair, 34 and $X1$, meets at C, and $X1$ is the chosen line, so C is known too. The theorem says the second pair, 23 and $5X$, meets on the line AC — so B is where 23 crosses AC, and $X$ is where the line from 5 through B crosses the chosen line. Every step is drawing a line through two known points or marking where two lines cross.

William Braikenridge and Colin Maclaurin found the construction independently in the 1730s, and it has a cleaner claim to fame than it is usually given: it draws a curved line with a tool that can only draw straight ones. Swing the chosen line through a half turn and the new point sweeps out the whole conic, which is how the thick curve in the figure is drawn — by running the construction at 719 directions. A straightedge by itself buys nothing among the constructible lengths; given five points on a conic, it buys the whole curve.

The figure does not trust the construction’s word for it. It solves for the conic through the five points as a six-coefficient equation — the orthogonal complement of five rows, by signed minors — and requires every point the straightedge produced to satisfy that equation. The two methods share nothing but the five points.

A curve the construction cannot name #

The construction does not know which kind of conic it is drawing, and it does not need to.

The type is decided by the sign of $B^2 - 4AC$, and the figure computes that sign from the solved coefficients to name the curve in its caption. The construction itself uses no coefficient. When the chosen line becomes parallel to an asymptote the fourth cut has no finite crossing, the new point is at infinity, and the trace breaks and resumes on the other branch. The straightedge treats an ellipse and a hyperbola identically; only a point at infinity tells them apart, which is the projective reading of the classification the discriminant made.

Points for lines, and lines for points #

Pascal’s theorem has a twin that was not found until 1806, by Charles Julien Brianchon, and it comes from exchanging the roles of points and lines.

A hexagon inscribed in a conic has its six corners on the curve. A hexagon circumscribed about one has its six sides tangent to the curve. Where Pascal’s theorem talks about the meeting points of sides, the twin talks about the lines joining corners; where Pascal’s conclusion is three points on a line, the twin’s is three lines through a point.

In homogeneous coordinates the exchange is literal. A point is a triple of numbers and so is a line, and “the point lies on the line” is the statement that their dot product is nought — symmetric in the two. The line through two points is their cross product, and so is the point where two lines meet. Every step of the proof of Pascal’s theorem can be read with the words swapped, and a conic has a dual conic — the set of its tangent lines — for the swapped proof to be about. Brianchon’s theorem is Pascal’s theorem, read in the other language. The figure checks each of its six lines for tangency by the condition $a^2u^2 + b^2v^2 = w^2$ on the line’s coefficients before building anything from them.

A broadside at sixteen #

Pascal published the theorem in 1640, at sixteen, as a single printed sheet headed Essai pour les coniques — a few hundred words and one figure, issued as a placard. He called the configuration the mystic hexagram and reportedly drew a great many consequences from it; the treatise on conics he built around them is lost, and what survives is the placard and the notes Gottfried Leibniz made from a manuscript decades later.

The setting explains the style. Pascal learned his geometry in the circle around Girard Desargues, whose book on conics of 1639 treated them as shadows of a circle and added points at infinity where parallel lines meet — the projective view in everything but name. It was almost entirely ignored for two centuries. Pascal’s theorem is exactly the kind of statement that view produces: it is about points and lines, it holds for every conic because it holds for one, and its proof sends a line to infinity.

Braikenridge’s and Maclaurin’s construction came a century later. Brianchon’s dual came in 1806, when geometers trained under Gaspard Monge in Paris were reviving the methods Desargues had started, and Jean-Victor Poncelet’s treatise of 1822 turned the exchange of points and lines from a coincidence noticed case by case into a principle. By then the hexagon was the standard example of a projective theorem, and its sixty lines a small subject of their own.

The theorem was found with the right idea two hundred years before the idea was accepted, which is why a result about arbitrary conics could be stated by a teenager without a single coordinate: it needed none.

Where the theorem needs its hypotheses #

The meeting points may be at infinity. Two opposite sides can be parallel, and then they do not meet in the plane. The theorem is true in the projective plane — the plane with a line at infinity added, on which parallel lines meet — and in that setting a Pascal line can be the line at infinity itself, as it is in the proof above. The figures choose their points so that all three meetings are finite and drawable, and refuse points that are not.

The six points must be distinct, or the sides must be read as tangents. Let two neighbouring corners slide together and the side between them becomes the tangent there. The theorem survives the limit, and the five-point and four-point versions that result are how a tangent to a conic is constructed with a straightedge.

The conic may be degenerate. If the “conic” is a pair of lines, with three of the six points on each, the theorem is still true and has its own name: it is Pappus’s theorem, from the fourth century, older than Pascal’s by thirteen hundred years.

And collinearity to twelve places is a measurement. The figures compute in floating point, so “nought” means below a threshold set far above the rounding error and far below any real failure. The control is what gives that threshold meaning, and the control for the crossed hexagon is why it moves every corner and not only the last.

One hexagon of sixty, and no line at infinity #

Each figure shows one hexagon of the sixty that its six points make, and one set of six points of the uncountably many on the curve. The theorem is about all of them; the figures are instances checked to high precision.

They also cannot show the projective argument. The proof sends a line to infinity, and a drawing of a line at infinity is not available; what the figures show is the finite situation before and after, with the meeting points kept on the page on purpose. And the curve in the straightedge figures is drawn from 719 runs of a construction that is about one line at a time — the continuity of the trace is evidence that the construction sweeps the curve, not a proof.

Still open: how many grid points avoid three in a line #

The fact under both of Pascal’s figures is that a line meets a conic in at most two points. It is also the fact behind the best answers to a question that is still open after more than a century.

Henry Dudeney asked in 1917 how many points can be chosen from an $n$ by $n$ grid with no three on a line. Each row can hold at most two, so the answer is at most $2n$, and computer searches have found arrangements reaching $2n$ for every grid up to about fifty on a side. For large grids nobody has come close. The best general constructions take the points of a conic in arithmetic modulo a prime — Paul Erdős used a parabola to get nearly $n$ points, and a 1975 construction by Richard Hall, Terence Jackson, Anthony Sudbery and Kenneth Wild used a hyperbola to get nearly $\tfrac32 n$ — and they work because a line meets a conic at most twice in that arithmetic too. Richard Guy and Patrick Kelly conjectured in 1968 that for large $n$ the true maximum is only about $1.81n$, short of $2n$. Neither that conjecture nor its opposite has been proved.

A theorem that measures nothing survives everything #

The habit is about reading a statement for what it uses.

Pascal’s theorem could have been stated with distances or angles or coefficients, and every such statement would have been true of one kind of conic at a time. Stated with points and lines alone, it is true of all four kinds, of crossed hexagons, of pairs of lines, and of tangents when corners merge — because every transformation that preserves incidence carries the theorem with it, and none of those transformations cares about length or angle. A statement that uses less survives more.

That is the same observation the classification made about its three unused coefficients, taken one level further. There, the discriminant ignored position and so could not be changed by moving the curve. Here, the theorem ignores measurement entirely and so cannot be changed by any projection. The test for how general a statement is, is the list of things it does not mention.