Other mathematical work includes a system he called Géométrographie and a method which related algebraic expressions to geometric objects.

During his tenure at the École Polytechnique and as a civil engineer, Lemoine published several papers on mathematics, most of which are included in a fourteen-page section in Nathan Altshiller Court's College Geometry.

Lemoine was born in Quimper, Finistère, on 22 November 1840, the son of a retired military captain who had participated in the campaigns of the First French Empire occurring after 1807.

As a child, he attended the military Prytanée of La Flèche on a scholarship granted because his father had helped found the school.

During this early period, he published a journal article in Nouvelles annales de mathématiques, discussing properties of the triangle.

[1] Lemoine was accepted into the École Polytechnique in Paris at the age of twenty, the same year as his father's death.

After graduation in 1866, he considered a career in law, but was discouraged by the fact that his advocacy for republican ideology and liberal religious views clashed with the ideals of the incumbent government, the Second French Empire.

[1] Lemoine also lectured at various scientific institutions in Paris and taught as a private tutor for a period before accepting an appointment as a professor at the École Polytechnique.

[1] As a founding member of the Association Française pour l'Avancement des Sciences, Lemoine presented what became his best-known paper, Note sur les propriétés du centre des médianes antiparallèles dans un triangle at the Association's 1874 meeting in Lille.

[1] Instead of the original idea, Lemoine proposed a simplification of the construction process to a number of basic operations with the compass and straightedge.

He published additional papers on the subject in Mathesis (1888), Journal des mathématiques élémentaires (1889), Nouvelles annales de mathématiques (1892), and the self-published La Géométrographie ou l'art des constructions géométriques, which was presented at the meeting of the Association Française in Pau (1892), and again at Besançon (1893) and Caen (1894).

[1] In 1894, Lemoine co-founded another mathematical journal entitled, L'intermédiaire des mathématiciens along with Charles Laisant, a friend whom he met at the École Polytechnique.

"[20] Lemoine has been described by Nathan Altshiller Court as a co-founder (along with Henri Brocard and Joseph Neuberg) of modern triangle geometry, a term used by William Gallatly, among others.

[21] Such geometry relies on the abstraction of figures in the plane rather than analytic methods used earlier involving specific angle measures and distances.