Lemoine's problem
In geometry, Lemoine's problem is a straightedge and compass construction problem posed by French mathematician Émile Lemoine in 1868:

Given one vertex of each of the equilateral triangles placed on the sides of a triangle, construct the original triangle. The problem was published as Question 864 in  (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  (series  2, Volume  8 (1869), pp 40–42) contained a description of a hyperbola which is now known as the Kiepert hyperbola. Ludwig Kiepert's solution

Kiepert establishes the validity of his construction by proving a few lemmas. Problem
Let  be the vertices of the equilateral triangles placed on the sides of a triangle  Given  construct . Lemma 1 
If on the three sides of an arbitrary triangle  one describes equilateral triangles    then the line segments    are equal, they concur in a point , and the angles they form one another are equal to 60°. Lemma 2
If on  one makes the same construction as that on  there  will have three equilateral triangles    three equal line segments    which will also concur at the point . Lemma 3
  are respectively the midpoints of   

Solution
Describe on the segments    the equilateral triangles    respectively. The midpoints of    are, respectively, the vertices  of the required triangle. Other solutions
Several other people in addition to Kiepert submitted their solutions  during 1868–9, including  Messrs Williere (at Arlon), Brocard, Claverie (Lycee de Clermont), Joffre (Lycee Charlemagne), Racine (Lycee de Poitiers), Augier (Lycee de Caen), V. Niebylowski, and L. Henri Lorrez. Kiepert's solution was more complete than the others. References

Triangle problems