![Malfatti's circles](https://upload.wikimedia.org/wikipedia/commons/3/3f/Malfatti%27s_circles.svg)
In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the Malfatti circles are three
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
s inside a given
triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, an ...
such that each circle is
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
to the other two and to two sides of the triangle. They are named after
Gian Francesco Malfatti
Giovanni Francesco Giuseppe Malfatti, also known as Gian Francesco or Gianfrancesco (26 September 1731 – 9 October 1807) was an Italian mathematician. He was born in Ala, Trentino, Italy and died in Ferrara.
Malfatti studied at the College of Sa ...
, who made early studies of the problem of constructing these circles in the mistaken belief that they would have the largest possible total area of any three disjoint circles within the triangle.
Malfatti's problem has been used to refer both to the problem of constructing the Malfatti circles and to the problem of finding three area-maximizing circles within a triangle.
A simple construction of the Malfatti circles was given by , and many mathematicians have since studied the problem. Malfatti himself supplied a formula for the radii of the three circles, and they may also be used to define two
triangle center
In geometry, a triangle center (or triangle centre) is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles, that is, a point that is in the middle of the figure by some measure. For example ...
s, the Ajima–Malfatti points of a triangle.
The problem of maximizing the total area of three circles in a triangle is never solved by the Malfatti circles. Instead, the optimal solution can always be found by a
greedy algorithm
A greedy algorithm is any algorithm that follows the problem-solving heuristic of making the locally optimal choice at each stage. In many problems, a greedy strategy does not produce an optimal solution, but a greedy heuristic can yield locally ...
that finds the largest circle within the given triangle, the largest circle within the three connected subsets of the triangle outside of the first circle, and the largest circle within the five connected subsets of the triangle outside of the first two circles. Although this procedure was first formulated in 1930, its correctness was not proven until 1994.
Malfatti's problem
![Malfatti circles in equilateral triangle](https://upload.wikimedia.org/wikipedia/commons/8/8e/Malfatti_circles_in_equilateral_triangle.svg)
posed the problem of cutting three cylindrical
column
A column or pillar in architecture and structural engineering is a structural element that transmits, through compression, the weight of the structure above to other structural elements below. In other words, a column is a compression member. ...
s out of a triangular prism of marble, maximizing the total volume of the columns. He assumed that the solution to this problem was given by three tangent circles within the triangular cross-section of the wedge. That is, more abstractly, he conjectured that the three Malfatti circles have the maximum total area of any three disjoint circles within a given triangle.
Malfatti's work was popularized for a wider readership in French by
Joseph Diaz Gergonne
Joseph Diez Gergonne (19 June 1771 at Nancy, France – 4 May 1859 at Montpellier, France) was a French mathematician and logician.
Life
In 1791, Gergonne enlisted in the French army as a captain. That army was undergoing rapid expansion becaus ...
in the first volume of his
''Annales'' (
1811
Events
January–March
* January 8 – An unsuccessful slave revolt is led by Charles Deslondes, in St. Charles and St. James Parishes, Louisiana.
* January 17 – Mexican War of Independence – Battle of Calderón Brid ...
), with further discussion in the second and tenth. However, Gergonne only stated the circle-tangency problem, not the area-maximizing one.
Malfatti's assumption that the two problems are equivalent is incorrect. , who went back to the original Italian text, observed that for some triangles a larger area can be achieved by a
greedy algorithm
A greedy algorithm is any algorithm that follows the problem-solving heuristic of making the locally optimal choice at each stage. In many problems, a greedy strategy does not produce an optimal solution, but a greedy heuristic can yield locally ...
that inscribes a single circle of maximal radius within the triangle, inscribes a second circle within one of the three remaining corners of the triangle, the one with the smallest angle, and inscribes a third circle within the largest of the five remaining pieces. The difference in area for an equilateral triangle is small, just over 1%, but as pointed out, for an
isosceles triangle
In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...
with a very sharp apex, the optimal circles (stacked one atop each other above the base of the triangle) have nearly twice the area of the Malfatti circles.
In fact, the Malfatti circles are never optimal. It was discovered through numerical computations in the 1960s, and later proven rigorously, that the Lob–Richmond procedure always produces the three circles with largest area, and that these are always larger than the Malfatti circles. conjectured more generally that, for any integer , the greedy algorithm finds the area-maximizing set of circles within a given triangle; the conjecture is known to be true for .
History
The problem of constructing three circles tangent to each other within a triangle was posed by the 18th-century Japanese mathematician
Ajima Naonobu
, also known as Ajima Manzō Chokuyen, was a Japanese mathematician of the Edo period.Smith, David. (1914).
His Dharma name was (祖眞院智算量空居士).
Work
Ajima is credited with introducing calculus into Japanese mathematics. The si ...
prior to the work of Malfatti, and included in an unpublished collection of Ajima's works made a year after Ajima's death by his student Kusaka Makoto. Even earlier, the same problem was considered in a 1384 manuscript by Gilio di Cecco da Montepulciano, now in the
Municipal Library of
Siena
Siena ( , ; lat, Sena Iulia) is a city in Tuscany, Italy. It is the capital of the province of Siena.
The city is historically linked to commercial and banking activities, having been a major banking center until the 13th and 14th centuri ...
,
Italy
Italy ( it, Italia ), officially the Italian Republic, ) or the Republic of Italy, is a country in Southern Europe. It is located in the middle of the Mediterranean Sea, and its territory largely coincides with the homonymous geographical re ...
. studied a special case of the problem, for a specific
isosceles triangle
In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...
.
Since the work of Malfatti, there has been a significant amount of work on methods for constructing Malfatti's three tangent circles;
Richard K. Guy
Richard Kenneth Guy (30 September 1916 – 9 March 2020) was a British mathematician. He was a professor in the Department of Mathematics at the University of Calgary. He is known for his work in number theory, geometry, recreational mathemati ...
writes that the literature on the problem is "extensive, widely scattered, and not always aware of itself".
Notably, presented a simple geometric construction based on
bitangent
In geometry, a bitangent to a curve is a line that touches in two distinct points and and that has the same direction as at these points. That is, is a tangent line at and at .
Bitangents of algebraic curves
In general, an algebraic cur ...
s; other authors have since claimed that Steiner's presentation lacked a proof, which was later supplied by , but Guy points to the proof scattered within two of Steiner's own papers from that time. Solutions based on algebraic formulations of the problem include those by , , , , and . The algebraic solutions do not distinguish between internal and external tangencies among the circles and the given triangle; if the problem is generalized to allow tangencies of either kind, then a given triangle will have 32 different solutions and conversely a triple of mutually tangent circles will be a solution for eight different triangles.
credits the enumeration of these solutions to , but notes that this count of the number of solutions was already given in a remark by . The problem and its generalizations were the subject of many other 19th-century mathematical publications, and its history and mathematics have been the subject of ongoing study since then.
It has also been a frequent topic in books on geometry.
and recount an episode in 19th-century
Neapolitan
Neapolitan means of or pertaining to Naples, a city in Italy; or to:
Geography and history
* Province of Naples, a province in the Campania region of southern Italy that includes the city
* Duchy of Naples, in existence during the Early and Hig ...
mathematics related to the Malfatti circles. In 1839,
Vincenzo Flauti
Vincenzo Flauti (1782–1863) was an Italian mathematician.
Life and work
Flauti studied at the ''Liceo del Salvatore'', the school led by Nicola Fergola. Although he began medical studies, he changed them to mathematics influenced by his ma ...
, a
synthetic geometer, posed a challenge involving the solution of three geometry problems, one of which was the construction of Malfatti's circles; his intention in doing so was to show the superiority of synthetic to analytic techniques. Despite a solution being given by Fortunato Padula, a student in a rival school of
analytic geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.
Analytic geometry is used in physics and engineerin ...
, Flauti awarded the prize to his own student, Nicola Trudi, whose solutions Flauti had known of when he posed his challenge. More recently, the problem of constructing the Malfatti circles has been used as a test problem for
computer algebra system
A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The de ...
s.
Steiner's construction
![Construction of Malfatti circles](https://upload.wikimedia.org/wikipedia/commons/2/2d/Construction_of_Malfatti_circles.svg)
Although much of the early work on the Malfatti circles used
analytic geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.
Analytic geometry is used in physics and engineerin ...
, provided the following simple
synthetic Synthetic things are composed of multiple parts, often with the implication that they are artificial. In particular, 'synthetic' may refer to:
Science
* Synthetic chemical or compound, produced by the process of chemical synthesis
* Synthetic o ...
construction.
A circle that is tangent to two sides of a triangle, as the Malfatti circles are, must be centered on one of the
angle bisector
In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a ''bisector''. The most often considered types of bisectors are the ''segment bisector'' (a line that passes through ...
s of the triangle (green in the figure). These bisectors partition the triangle into three smaller triangles, and Steiner's construction of the Malfatti circles begins by drawing a different triple of circles (shown dashed in the figure) inscribed within each of these three smaller triangles. In general these circles are disjoint, so each pair of two circles has four
bitangent
In geometry, a bitangent to a curve is a line that touches in two distinct points and and that has the same direction as at these points. That is, is a tangent line at and at .
Bitangents of algebraic curves
In general, an algebraic cur ...
s (lines touching both). Two of these bitangents pass ''between'' their circles: one is an angle bisector, and the second is shown as a red dashed line in the figure. Label the three sides of the given triangle as , , and , and label the three bitangents that are not angle bisectors as , , and , where is the bitangent to the two circles that do not touch side , is the bitangent to the two circles that do not touch side , and is the bitangent to the two circles that do not touch side . Then the three Malfatti circles are the inscribed circles to the three
tangential quadrilateral
In Euclidean geometry, a tangential quadrilateral (sometimes just tangent quadrilateral) or circumscribed quadrilateral is a convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This circle is called the ...
s , , and . In case of symmetry two of the dashed circles may touch in a point on a bisector, making two bitangents coincide there, but still setting up the relevant quadrilaterals for Malfatti's circles.
The three bitangents , , and cross the triangle sides at the point of tangency with the third inscribed circle, and may also be found as the reflections of the angle bisectors across the lines connecting pairs of centers of these incircles.
[.]
Radius formula
The
radius
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
of each of the three Malfatti circles may be determined as a formula involving the three side lengths , , and of the triangle, the
inradius
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter.
...
, the
semiperimeter
In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate name ...
, and the three distances , , and from the
incenter
In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisec ...
of the triangle to the vertices opposite sides , , and respectively. The formulae for the three radii are:
Related formulae may be used to find examples of triangles whose side lengths, inradii, and Malfatti radii are all
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s or all integers. For instance, the triangle with side lengths 28392, 21000, and 25872 has inradius 6930 and Malfatti radii 3969, 4900, and 4356. As another example, the triangle with side lengths 152460, 165000, and 190740 has inradius 47520 and Malfatti radii 27225, 30976, and 32400.
Ajima–Malfatti points
![Primo punto di Malfatti](https://upload.wikimedia.org/wikipedia/commons/a/ad/Primo_punto_di_Malfatti.svg)
Given a triangle ''ABC'' and its three Malfatti circles, let ''D'', ''E'', and ''F'' be the points where two of the circles touch each other, opposite vertices ''A'', ''B'', and ''C'' respectively. Then the three lines ''AD'', ''BE'', and ''CF'' meet in a single
triangle center
In geometry, a triangle center (or triangle centre) is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles, that is, a point that is in the middle of the figure by some measure. For example ...
known as the first Ajima–Malfatti point after the contributions of Ajima and Malfatti to the circle problem. The second Ajima–Malfatti point is the meeting point of three lines connecting the tangencies of the Malfatti circles with the centers of the
excircle
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter.
...
s of the triangle. Other triangle centers also associated with the Malfatti circles include the Yff–Malfatti point, formed in the same way as the first Malfatti point from three mutually tangent circles that are all tangent to the lines through the sides of the given triangle, but that lie partially outside the triangle, and the
radical center
The term radical center can refer to:
* Radical centrism, a political movement
* a mathematical construct: also called the power center (geometry)
{{Disambig ...
of the three Malfatti circles (the point where the three bitangents used in their construction meet).
[.]
See also
*
Circle packing in an equilateral triangle
Circle packing in an equilateral triangle is a packing problem in discrete mathematics where the objective is to pack unit circles into the smallest possible equilateral triangle. Optimal solutions are known for and for any triangular number o ...
*
Circle packing in an isosceles right triangle
Circle packing in a right isosceles triangle is a packing problem where the objective is to pack unit circles into the smallest possible isosceles right triangle.
Minimum solutions (lengths shown are length of leg) are shown in the table below. ...
*
Six circles theorem
In geometry, the six circles theorem relates to a chain of six circles together with a triangle, such that each circle is tangent to two sides of the triangle and also to the preceding circle in the chain. The chain closes, in the sense that the ...
Notes
References
*.
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*. Continued in vol. 11 (1896)
pp. 25–27
*
*.
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*. Reprinted in .
*. Reprinted in .
*. Reprinted in .
*. Reprinted in .
*.
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*. See also .
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*. The cover of Martin's book features an illustration of the Malfatti circles.
*.
*.
*.
*. Proposed by
Artemas Martin
Artemas Martin (August 3, 1835 – November 7, 1918) was a self-educated American mathematician.
Biography
Martin was born on August 3, 1835, in Steuben County, New York, grew up in Venango County, Pennsylvania, and spent most of his life in E ...
; solved by the proposer and by Asher B. Evans; compare Martin's Question 4401, also in this volume, pp. 102–103, again solved by Evans and Martin. Note further that Martin had asked for a geometrical solution i
''The Lady's and Gentleman's Diary''for 1869 (so appearing in late 1868), with solution in the LDG for the following year, pp. 89–90. Versions of the problem then appear from 1879 in ''The Mathematical Visitor'', edited by Martin.
*
*.
*.
*.
*.
*.
*. As cited by and .
*.
*.
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*. As cited by .
*.
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*.
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*. Reprinted in and separately as . See in particular section 14
pp. 25–27 of the Engelmann reprint
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*. See also .
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External links
* {{mathworld2, urlname=MalfattiCircles, title=Malfatti Circles, urlname2=MalfattisProblem, title2=Malfatti's Problem
Malfatti's Problemat
cut-the-knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...
Circle packing
Circles defined for a triangle