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Proving the solution of Malfatti’s marble problem

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Rendiconti del Circolo Matematico di Palermo Series 2 Aims and scope Submit manuscript

Mή μου τοὺς κύκλους τάραττε

’Αρχιμήδης ‘ο Συρακóσιος

Noli turbare circulos meos

Archimedes Syracusanus

Abstract

Malfatti’s problem is more than 200 years old and its solution given by the greedy arrangement has been stated first in 1994. The solution was obtained at that time by considering 14 arrangements and by justifying the exclusion of the non-greedy ones through a complex reasoning relying purely on numerical simulations in several crucial steps. By significantly improving the underlying methods and by filling still extant gaps, a full purely analytical proof of the solution is provided for the first time in the present work, especially based on a mixture of synthetic and analytic Euclidean geometry and on the theory of convex functions.

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Notes

  1. Any other larger circle would have the center interior to one of the three triangles obtained joining the in-center with the vertices of the triangle and then would overlap at least the corresponding L of the LLL triangle.

  2. Another known construction is to consider the center C3 of C and to inscribe a given arbitrary circle in the angle with parallel sides to LL and vertex V4 in Fig. 2a [7], wherein the sides of this last angle are at a distance equal to the radius of C from the sides of the angle LL; the ray V4C3 always intersects the given circle in two points and the two solutions follow by homothety centered in V4.

  3. This fact may be shown also by inversion in the contact point of CC, transforming CC in two parallel lines and L in a circle tangent to them. For sake of completeness, the same reasoning applies to the curvilinear triangle delimited by tangent CCC, thus also admitting an in-circle: the in-center is the common intersection of three hyperbola branches with respective vertices at the CCC contact points. One may see by reasoning as in the LLL case that the in-circle is maximum among the circles contained in the triangle, which is then a rigidity zone.

  4. The center of the maximal circle inscribed in a given angle of a quadrangle is obtained by taking the bisectors of the adjacent two angles, intersecting them with the bisector of the angle and taking the intersection closest to the vertex.

  5. If C ≠ E, the side AD is a common external tangent of the maximal circles and the maximal circle in ∠D touches BC, whereas if C = E, two opposite sides of the quadrangle are common external tangents; then with respect to a common internal tangent not containing B, B and D lie on opposite sides, as well as the circles, and every couple of smaller circles inscribed respectively in the angles is a fortiori separated by this tangent.

  6. t sinα0(t) < 1, and √R3m(t) are positive increasing, whereas sinα0(t)/t < 1, and 1/cosα0(t) are positive decreasing for t in (5).

  7. The singularity of γ0´(t) at t = t2 hinders analysis of the function α0 + β0 + γ0: by numerical computation one could check that this value is also unique in 1 < t < t2, but this is unnecessary for the development of the proof.

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  1. Munich, Germany

    Giancarlo Lombardi

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  1. Giancarlo Lombardi
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Lombardi, G. Proving the solution of Malfatti’s marble problem. Rend. Circ. Mat. Palermo, II. Ser 72, 1751–1782 (2023). https://doi.org/10.1007/s12215-022-00759-2

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