Other results in the paper included the idea that the symmedian from a vertex of the triangle divides the opposite side into segments whose ratio is equal to the ratio of the squares of the other two sides. Lemoine also proved that if lines are drawn through the Lemoine point parallel to the sides of the triangle, then the six points of intersection of the lines and the sides of the triangle are concyclic, or that they lie on a circle. This circle is now known as the first Lemoine circle, or simply the Lemoine circle. Construction system Lemoine's system of constructions, the Géométrographie, attempted to create a methodological system by which constructions could be judged. This system enabled a more direct process for simplifying existing constructions. In his description, he listed five main operations: placing a compass's end on a given point, placing it on a given line, drawing a circle with the compass placed upon the aforementioned point or line, placing a straightedge on a given line, and extending the line with the straightedge. The "simplicity" of a construction could be measured by the number of its operations. In his paper, he discussed as an example the Apollonius problem originally posed by Apollonius of Perga during the Hellenistic period; the method of constructing a circle tangent to three given circles. The problem had already been solved by Joseph Diaz Gergonne in 1816 with a construction of simplicity 400, but Lemoine's presented solution had simplicity 154. Simpler solutions such as those by Frederick Soddy in 1936 and by David Eppstein in 2001 are now known to exist. Lemoine's conjecture and extensions In 1894, Lemoine stated what is now known as Lemoine's conjecture: Every odd number which is greater than three can be expressed in the form 2p + q where p and q are prime. In 1985, John Kiltinen and Peter Young conjectured an extension of the conjecture which they called the "refined Lemoine conjecture". They published the conjecture in a journal of the Mathematical Association of America: "For any odd number m which is at least 9, there are odd prime numbers p, q, r and s and positive integers j and k such that m = 2p + q, 2 + pq = 2j + r and 2q + p = 2k + s. [...] the study has directed our attention to more subtle aspects of the