Pappus chain
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Q.
Starting with the circle P1 tangent to the three semicircles that form an arbelos, construct a sequence of circles Pi, each tangent to one of the two smaller inner circles and the large outer circle. This chain of circles is called the Pappus chain. In the Pappus chain, the distance between the center of the first inscribed circle P1 and the baseline is twice the radius of the circle, and the distance between the center of the second circle P2 and the baseline is four times its radius. In general, the distance between the center of the nth circle Pn and the baseline is 2n times its radius. Furthermore, the centers of the circles Pi lie on a single ellipse.
Solution.
First, let us introduce the following notation.
| Object | Center | Radius | Height of Center |
|---|---|---|---|
| Large outer semicircle | $C$ | $R$ | $0$ |
| Left inner semicircle | $L$ | $r_L$ | $0$ |
| Right inner semicircle | $M$ | $r_M$ | $0$ |
| $n$-th circle in the chain | $P_n$ | $\rho_n$ | $h_n$ |
CPn = R - ρn, LPn = rL + ρn
∴ CPn + LPn = R + rL
Therefore, the centers of the circles forming the Pappus chain lie on an ellipse with C and L as its two foci.
To show that the height of the center of Pn is 2rn, let us use inversive geometry with respect to point A. The inversion of a point P with respect to a reference circle centered at A with radius k is the point P’, satisfying AP × AP’ = k2, with points A, P, and P’ lying on a single straight line. Letting AP = 2R* cos θ gives AP’ = (k2 / 2R) / cos θ, so the inverse image of a circle passing through A is a line perpendicular to the x-axis and passing through x = k2 / 2R. Therefore, in the figure above, we can identify two parallel lines passing through x = k2 / 2R and x = k2 / 2rL.
By the principle of continuity, the inverse image of the inner circle on the right is still a circle. Therefore, the inverse images of the 1st, 2nd, ⋯, nth, ⋯ chain circles are also circles, and by the principle of continuity, they are transformed into congruent circles of radius σ tangent between the two parallel lines. Since the congruent circles are stacked vertically at intervals equal to their diameter 2σ, the height of the inverse image of the nth chain circle is Hn = 2nσ. If the position of the center and the radius of the inverted circle are scaled by a factor of λn, then by similarity, the height of the inverted circle is also scaled by a factor of λn. Therefore, 2n = 2nσ / σ = Hn / σ = λnhn / λnρn = hn / ρn ⇔ hn = 2nρn.
Meanwhile, defining r = rL / R, the original proof shows by mathematical induction that Pn = (xn, yn) and ρn = rn.
Input: 2026.08.16 18:32