One number for every chord through a point

An angle that does not care where it stands found that a chord of a circle is seen at the same angle from every point of the arc opposite it. That essay was about angles. This one is about lengths, and it shows that the constant angle forces a constant product: the first place where the inscribed angle stops being a fact about angles and starts measuring distances.

Take a point $P$ inside a circle and draw a straight line through it. The line meets the circle twice, at $A$ and $B$, and cuts the chord into two pieces, $PA$ and $PB$. Draw another line through $P$, and another. The pieces change length completely — a chord nearly through the centre is split nearly evenly, a short chord near the edge very unevenly — and yet the product $PA \cdot PB$ is the same for every line through $P$.

The figure’s point sits 3 units from the centre of a circle of radius 5. The horizontal chord through it is split into 2 and 8; the others into 7.01 and 2.28, 4.67 and 3.42, 2.71 and 5.89. Every product is 16. The rectangles on the right make the claim visible: they are drawn with the two pieces of each chord as their sides, on one scale, and they are four quite different shapes with exactly the same area.

Why the product cannot change #

Take two chords through $P$, say $AB$ and $CD$, and join $A$ to $C$ and $D$ to $B$. That makes two triangles, $PAC$ and $PDB$, sharing the vertex $P$.

The angle at $A$ in the first triangle, between $AP$ and $AC$, is the angle at which $A$ sees the chord $CB$ — since $AP$ runs along $AB$. The angle at $D$ in the second triangle is the angle at which $D$ sees the same chord $CB$. By the theorem of the earlier essay, two points on the same arc see a chord at the same angle, so these two angles are equal. The same argument gives equal angles at $C$ and $B$, and the angles at $P$ are vertically opposite. The two triangles are similar.

Similar triangles have proportional sides: $PA/PD = PC/PB$. Cross-multiplying gives $PA \cdot PB = PC \cdot PD$. Any two chords through $P$ give the same product, so all of them do. The inscribed angle has done all the work; without it there is no reason for the two triangles to have any angles in common.

The number itself comes from the chord through the centre. It is split into $r - d$ and $r + d$, where $r$ is the radius and $d$ the distance from $P$ to the centre, so the product is $(r - d)(r + d) = r^2 - d^2$. In the figure, $25 - 9 = 16$. A point at the centre has the largest product, $r^2$, and a point near the circle a product close to nought, because one of its two pieces is always short.

The same fact, as the product of two roots #

There is a second proof that makes no use of angles at all, and it explains why it is the product and not the sum that stays fixed. Put the centre of the circle at the origin and let a line leave $P$ in the direction of a unit vector $u$. Its points are $P + tu$, and such a point is on the circle when $|P + tu|^2 = r^2$, which expands to a quadratic in $t$:

$$t^2 + 2(P \cdot u)\,t + \bigl(|P|^2 - r^2\bigr) = 0.$$

The two roots $t_1$ and $t_2$ are the signed distances from $P$ to the two crossing points. Their sum is $-2(P\cdot u)$, which depends on the direction $u$: a chord through $P$ towards the centre is lopsided one way, a chord away from it the other. Their product is the constant term, $|P|^2 - r^2$, which does not mention $u$ at all. This is what the coefficients already know applied to one quadratic: the product of the roots is read off the constant term, and here the constant term happens to be the only part of the equation that does not care which way the line points.

The two proofs are the same fact seen from two sides. The similar triangles explain it geometrically, with the inscribed angle; the quadratic explains it algebraically, with the direction dropping out of the last coefficient. Neither needs the other, and they agree on the number.

From outside, and along a tangent #

Move $P$ outside the circle. A line through it now meets the circle at a near point and a far point, both on the same side of $P$, and the same similar-triangle argument — with the triangles now overlapping rather than on opposite sides of $P$ — shows that near distance times far distance is again the same for every line.

The chord through the centre gives the value: near distance $d - r$, far distance $d + r$, product $d^2 - r^2$. In the figure that is $169 - 25 = 144$. Now tilt the line until it only grazes the circle. The near and far points slide together and meet at the point of tangency, so both distances become the tangent length $t$, and the product becomes $t^2$. Hence $t^2 = 144$ and $t = 12$ — which is also Pythagoras, since the tangent meets the radius at a right angle and $12^2 + 5^2 = 13^2$. The secant theorem and the tangent’s length are one fact, with the tangent as the limiting secant.

Euclid had both, as Propositions 35 and 36 of the third book of the Elements: the chords through an interior point, and the secant and tangent from an exterior one. They look like two theorems because the pictures look different. They are one theorem about one number.

One signed number per point #

The two cases can be made into a single formula by counting direction. Measure $PA$ and $PB$ along the line with a sign — positive in one direction, negative in the other. Inside the circle, $A$ and $B$ lie on opposite sides of $P$, so the signed product is negative; outside, on the same side, it is positive; on the circle one of them is zero.

With that convention the product is $d^2 - r^2$ everywhere — a single number attached to each point of the plane, called the power of the point with respect to the circle. Jacob Steiner gave it the name in 1826. The figure measures it at forty-one points along a line, each time with a chord in a randomly chosen direction, and every measurement lands on the parabola $d^2 - 25$. The random directions are the point of the experiment: whichever way the line through a point is drawn, the product it reports is the same.

In coordinates the power is simply the circle’s own equation, evaluated off the circle. A circle is the set of points where $x^2 + y^2 + Dx + Ey + F = 0$; put any point into the left-hand side and the number that comes out is that point’s power. The equation of a circle is usually read as a test — zero means on the circle — and it is also a measurement, telling each point of the plane how far inside or outside it lies in exactly the units of the chord products.

The power is what makes the circle a level set: the circle is exactly where the power is zero. Two circles give two powers, and the points where they are equal form a straight line, the radical axis — straight because subtracting one circle’s equation from the other’s cancels the $x^2 + y^2$ they share and leaves an equation of the first degree; it is the boundary between the cells of a power diagram, where sites of different weights divide the plane, and three circles’ radical axes meet at a point used in eight circles touching three. Those are the power’s uses. Its origin is the pair of similar triangles above.

Four points on a circle, tested without the circle #

The theorem runs backwards. Suppose two lines cross at $P$, with points $A$ and $B$ on one and $C$ and $D$ on the other. If the four points lie on a circle, the products agree. The converse is also true: if $PA \cdot PB = PC \cdot PD$, with signs, then $A$, $B$, $C$ and $D$ lie on one circle.

The proof is short. Draw the circle through $A$, $B$ and $C$; the line through $P$ and $C$ meets it again at some point $D'$ with $PA \cdot PB = PC \cdot PD'$. If also $PA \cdot PB = PC \cdot PD$, then $PD = PD'$, and $D$ and $D'$ are the same point. So the product test decides concyclicity without drawing a circle or measuring an angle, and it works equally well with $P$ inside the four points or outside them.

This is often the fastest way to prove that four points in a figure lie on a circle — find two lines through a common point and compare products — and it is the metric counterpart of the angle test from the earlier essay, where four points are concyclic when two of them see the segment between the other two at equal angles.

Every square root a compass makes #

The case with the most consequences is the most symmetric one. Take a diameter of a circle and a point $P$ on it, splitting it into pieces $a$ and $b$. The chord through $P$ perpendicular to the diameter is cut in half at $P$ — the circle is symmetric about the diameter — so both of its pieces have the same length $h$. The theorem says $h \cdot h = a \cdot b$.

So $h = \sqrt{ab}$, the geometric mean of $a$ and $b$, drawn with one circle and one perpendicular. Take $a = 1$ and the half-chord is $\sqrt b$: a square root of any length, constructed. This is Euclid’s construction of a square equal to a given rectangle, and it is the single operation behind the whole theory of constructible numbers. Every step is a square root found that compass-and-straightedge constructions reach exactly the numbers obtained from $1$ by arithmetic and square roots; the square roots come from exactly this half-chord, and the intersecting chords theorem is why its length is what it is. The circles in what two points can build are square roots waiting to be read off.

The same picture is a right-angled triangle in disguise. Join the top of the half-chord to the two ends of the diameter: by Thales the angle there is a right angle, and the half-chord is the altitude onto the hypotenuse, so the altitude of a right triangle is the geometric mean of the two pieces it cuts the hypotenuse into. The legs are geometric means too, by the tangent form of the theorem. The circle drawn on the other leg as its diameter passes through the foot of the altitude, since the angle there is a right angle; the first leg meets that circle at a right angle to its diameter, so it is a tangent to it; and the hypotenuse, from the same corner, is a secant through the foot and the far corner. The power of that corner then gives the leg next to the piece $a$ a square of $a(a + b)$, and the same argument at the other corner gives the other leg a square of $b(a + b)$. Adding, the squares of the legs sum to $(a + b)^2$, the square of the hypotenuse. The power of a point proves Pythagoras, in two lines, with no rearrangement of any area.

The same picture proves the inequality between the two means. The half-chord can be no longer than the radius, which is $(a + b)/2$, so $\sqrt{ab} \le (a + b)/2$, with equality only when the split is at the centre. The geometric mean of two lengths never exceeds their arithmetic mean, and the proof is that a half-chord is never longer than a radius.

Where inversion comes from #

The power also underlies the map in the map that trades circles for lines. Inversion in a circle of radius $k$ about $O$ sends a point at distance $d$ to the point on the same ray at distance $k^2/d$. Take a circle that does not pass through $O$ and consider the lines from $O$ through it: each meets it at a near and a far point, and by the secant theorem their distances multiply to the power of $O$, a fixed number $p$. Inversion multiplies the near distance’s reciprocal by $k^2$, so it sends the near point to distance $k^2/d_{\text{near}} = (k^2/p)\,d_{\text{far}}$ — a fixed multiple of the far point’s distance. The image of the circle is the circle scaled from $O$ by $k^2/p$. Circles invert to circles because of the power of a point, and when $k^2 = p$ the circle inverts to itself, which is the condition the eight-circles essay used to move three circles at once.

What the rectangles cannot show #

That the product is exactly equal, rather than nearly. The rectangles in the first figure are drawn to scale and look equal, and the printed products agree to two decimal places. Equality is the similar-triangle argument, which holds exactly for every chord; the pictures confirm four chords and the sweep forty-one directions, and none of that is the proof.

The sign. The rectangles have positive area, and the inside and outside cases are drawn in separate figures. The signed version, in which one formula covers both, is visible only in the sweep, and even there the sign is a convention chosen to make the formula uniform. That the convention is the right one — that it makes the radical axis a straight line and inversion a map of circles — is an argument, not a picture.

Why the angle forces the length. The similar triangles are drawn with their equal angles marked. What makes the angles equal is the inscribed angle theorem, which the drawing assumes. A reader who has not seen that theorem sees two triangles that happen to look alike.

Still open: how often a distance can repeat #

The power of a point measures distances from one point to a circle. A much older question asks how often a single distance can occur among many points. Place $n$ points in the plane and count the pairs exactly one unit apart. Paul Erdős asked in 1946 how large that count can be, and found that points of a suitably chosen square grid give slightly more than $n$ times a constant — about $n^{1 + c/\log\log n}$.

The best upper bound, proved by Joel Spencer, Endre Szemerédi and William Trotter in 1984, is of order $n^{4/3}$, and its proof counts how often points can lie on unit circles centred at other points — incidences between points and circles, the counting cousin of the concyclicity test above. Between the two bounds, the true answer is not known, and it has not moved in forty years. Whether many points can crowd onto many small circles at once is exactly the question, and no argument yet decides it.

A number that belongs to the point #

Every line through a point cuts a circle into two pieces whose lengths multiply to the same number, because two such lines make two triangles whose angles the inscribed angle theorem forces to match. Inside the circle the number is $r^2 - d^2$, outside it is $d^2 - r^2$ and is the square of the tangent, and with signs it is one quantity, the power.

Run backwards, the equal products decide when four points lie on a circle. Made symmetric, they draw a square root with one circle — the operation every compass-and-straightedge construction is made of. And sent through inversion, they are the reason circles invert to circles: a fact about lengths that began as a fact about angles.