Lemoine's problem

In mathematics, Lemoine's problem is a certain construction problem in elementary plane geometry posed by the French mathematician Émile Lemoine (1840–1912) in 1868.^[1][2] The problem was published as Question 864 in Nouvelles Annales de Mathématiques (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 published in Nouvelles Annales de Mathématiques (series 2, Volume 8 (1869), pp 40–42) contained a description of a hyperbola which is now known as the Kiepert hyperbola.^[3]

Statement of the problem

The question published by Lemoine poses the following construction problem:

Given the vertices of the equilateral triangles placed on the sides of a triangle, construct the original triangle.

Ludwig Kiepert's solution

Diagram illustrating Lemma 1 .

Diagram illustrating Ludwig Kiepert's solution to Lemoine's problem

Kiepert establishes the validity of his construction by proving a few lemmas.^[3][4]

Problem

Let A₁, B₁, C₁ be the vertices of the equilateral triangles placed on the sides of a triangle ABC. Given A₁, B₁, C₁ construct A, B, C.

Lemma 1

If on the three sides of an arbitrary triangle ABC, one describes equilateral triangles ABC₁, ACB₁, BCA₁, then the line segments AA₁, BB₁, CC₁ are equal, they concur in a point P, and the angles they form one another are equal to 60°.

Lemma 2

If on A₁B₁C₁ one makes the same construction as that on ABC, there will have three equilateral triangles A₁B₁C₂, A₁C₁B₂, B₁C₁A₂, three equal line segments A₁A₂, B₁B₂, C₁C₂, which will also concur at the point P.

Lemma 3

A, B, C are respectively the midpoints of A₁A₂, B₁B₂, C₁C₂.

Solution

• Describe on the segments A₁B₁, A₁C₁, B₁C₁ the equilateral triangles A₁B₁C₂, A₁C₁B₂, B₁C₁A₂, respectively.
• The midpoints of A₁A₂, B₁B₂, C₁C₂ are, respectively, the vertices A, B, C of the required triangle.

Other solutions

Kiepert was not the only person to submit solutions to the problem. There were several others who submitted their solutions during 1868–9. Those who submitted solutions include Messrs Williere (at Arlon), Brocard, Claverie (Lycee de Clermont), Joffre (Lycee Charlomagne), Racine (Lycee de Poitiers), Augier (Lycee de Caen), V. Niebylowski, L. Henri Lorrez. Kiepert's solution was more complete than the others.^[3]

References