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 ⁠${\displaystyle \triangle ABC.}$⁠ Given A₁, B₁, C₁ construct A, B, C.

Lemma 1

If on the three sides of an arbitrary triangle ⁠${\displaystyle \triangle ABC,}$⁠ one describes equilateral triangles ⁠${\displaystyle \triangle ABC_{1},}$⁠ ⁠${\displaystyle \triangle ACB_{1},}$⁠ ⁠${\displaystyle \triangle BCA_{1},}$⁠ then the line segments ⁠${\displaystyle {\overline {AA_{1}}},}$⁠ ⁠${\displaystyle {\overline {BB_{1}}},}$⁠ ⁠${\displaystyle {\overline {CC_{1}}}}$⁠ are equal, they concur in a point P, and the angles they form one another are equal to 60°.

Lemma 2

If on ⁠${\displaystyle \triangle A_{1}B_{1}C_{1}}$⁠ one makes the same construction as that on ⁠${\displaystyle \triangle ABC,}$⁠ there will have three equilateral triangles ⁠${\displaystyle \triangle A_{1}B_{1}C_{2},}$⁠ ⁠${\displaystyle \triangle A_{1}C_{1}B_{2},}$⁠ ⁠${\displaystyle \triangle B_{1}C_{1}A_{2},}$⁠ three equal line segments ⁠${\displaystyle {\overline {A_{1}A_{2}}},}$⁠ ⁠${\displaystyle {\overline {B_{2}B_{2}}},}$⁠ ⁠${\displaystyle {\overline {C_{2}C_{2}}},}$⁠ which will also concur at the point P.

Lemma 3

A, B, C are respectively the midpoints of ⁠${\displaystyle {\overline {A_{1}A_{2}}},}$⁠ ⁠${\displaystyle {\overline {B_{2}B_{2}}},}$⁠ ⁠${\displaystyle {\overline {C_{2}C_{2}}}.}$⁠

Solution

• Describe on the segments ⁠${\displaystyle {\overline {A_{1}B_{1}}},}$⁠ ⁠${\displaystyle {\overline {A_{1}C_{1}}},}$⁠ ⁠${\displaystyle {\overline {B_{1}C_{1}}}}$⁠ the equilateral triangles ⁠${\displaystyle \triangle A_{1}B_{1}C_{1},}$⁠ ⁠${\displaystyle \triangle A_{1}C_{2}B_{2},}$⁠ ⁠${\displaystyle \triangle B_{1}C_{1}A_{2},}$⁠ respectively.
• The midpoints of ⁠${\displaystyle {\overline {A_{1}A_{2}}},}$⁠ ⁠${\displaystyle {\overline {B_{2}B_{2}}},}$⁠ ⁠${\displaystyle {\overline {C_{2}C_{2}}}}$⁠ are, respectively, the vertices A, B, C of the required triangle.

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.^[3]