mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, Lemoine's problem is a certain construction problem in elementary

plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * Planes (gen ...

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 ...

posed by the

French French (french: français(e), link=no) may refer to: * Something of, from, or related to France ** French language, which originated in France, and its various dialects and accents ** French people, a nation and ethnic group identified with Franc ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

Émile Lemoine Émile Michel Hyacinthe Lemoine (; 22 November 1840 – 21 February 1912) was a French civil engineer and a mathematician, a geometer in particular. He was educated at a variety of institutions, including the Prytanée National Militaire and, most ...

(1840–1912) in 1868. The problem was published as Question 864 in ''

Nouvelles Annales de Mathématiques The ''Nouvelles Annales de Mathématiques'' (subtitled ''Journal des candidats aux écoles polytechnique et normale'') was a French scientific journal in mathematics. It was established in 1842 by Olry Terquem and Camille-Christophe Gerono, and con ...

'' (Series 2, Volume 7 (1868), p 191). The chief interest in the problem is that a discussion of the solution of the problem by

Ludwig Kiepert Friedrich Wilhelm August Ludwig Kiepert (6 October 1846 – 5 September 1934) was a German mathematician who introduced the Kiepert hyperbola. Selected works ''De curvis quarum arcus integralibus ellipticis primi generis exprimuntur'' 1870, dis ...

published in ''Nouvelles Annales de Mathématiques'' (series 2, Volume 8 (1869), pp 40–42) contained a description of a

hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, cal ...

which is now known as the Kiepert hyperbola.

Statement of the problem

The question published by Lemoine poses the following construction problem: :''Given one

vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...

of each of the

equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...

s placed on the sides of a

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 ...

, construct the original triangle.''

Ludwig Kiepert's solution

Kiepert establishes the validity of his construction by proving a few

lemma Lemma may refer to: Language and linguistics * Lemma (morphology), the canonical, dictionary or citation form of a word * Lemma (psycholinguistics), a mental abstraction of a word about to be uttered Science and mathematics * Lemma (botany), a ...

s.The details of the construction as given by Kiepert in French can be read here

/ref> :Problem :Let ''A''₁, ''B''₁, ''C''₁ be the vertices of the

s placed on the sides of a

''ABC''. Given ''A''₁, ''B''₁, ''C''₁ construct ''A'', ''B'', ''C''. :Lemma 1 :If on the three sides of an arbitrary triangle ''ABC'', one describes equilateral triangles ''ABC''₁, ''ACB''₁, ''BCA''₁, then the line segments ''AA''₁, ''BB''₁, ''C''C₁ are equal, they concurrent lines, concur in a point ''P'', and the angles they form one another are equal to 60°. :Lemma 2 :If on ''A''₁''B''₁''C''₁ one makes the same construction as that on ''ABC'', there will have three equilateral triangles ''A''₁''B''₁''C''₂, ''A''₁''C''₁''B''₂, ''B''₁''C''₁''A''₂, three equal line segments ''A''₁''A''₂, ''B''₁''B''₂, ''C''₁''C''₂, which will also concur at the point ''P''. :Lemma 3 : ''A'', ''B'', ''C'' are respectively the

midpoint In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. Formula The midpoint of a segment in ''n''-dimens ...

s of ''A''₁''A''₂, ''B''₁''B''₂, ''C''₁''C''₂. :Solution :*Describe on the segments ''A''₁''B''₁, ''A''₁''C''₁, ''B''₁''C''₁ the equilateral triangles ''A''₁''B''₁''C''₂, ''A''₁''C''₁''B''₂, ''B''₁''C''₁''A''₂, respectively. :*The midpoints of ''A''₁''A''₂, ''B''₁''B''₂, ''C''₁''C''₂ are, respectively, the vertices ''A'', ''B'', ''C'' of the required triangle.

References

{{reflist Triangle problems

Statement of the problem

Ludwig Kiepert's solution

Other solutions

References