At the École Polytechnique in Paris, Brianchon studied under Monge. In fact he published his first paper Sur les surfaces courbes du second degré Ⓣ in the Journal de l'École Polytechnique while still a student. In that paper Brianchon rediscovered Pascal's Magic Hexagon. He showed that in any hexagon formed of six tangents to a conic, the three diagonals meet at a point. This result is often called Brianchon's Theorem and it is the result for which he is best known. In fact this theorem is simply the dual of Pascal's theorem which was proved in 1639:-
If all the vertices of a hexagon lie on a conic, and if the opposite sides intersect, then the points of intersection lie on a line.In  Greitzer points out that Pascal recognised that his theorem was projective in nature so it is surprising that it took 167 years before someone realised that its dual, which is Brianchon's Theorem, would also be true.
Brianchon graduated in 1808 as first in his class. One might have expected him to continue at this stage to an academic career but these were unusual times in France. Napoleon Bonaparte had declared an empire in 1804 with himself as Emperor. He basically controlled continental Europe and he had only the British to fight against, but without control of the seas he could not mount an invasion. The British under Nelson won a decisive victory at Trafalgar where the Franco-Spanish fleet was destroyed. Napoleon then tried the tactic of blockading Britain and so he gave an order to stop all trade with the British Isles. However Portugal was reluctant to stop trading with Britain, both for economic and political reasons, and Napoleon decided to send his armies to Portugal to force them to comply with his orders. At around this time Brianchon graduated from the École Polytechnique and became a lieutenant in artillery in Napoleon's army.
Although Spain allowed Napoleon's armies to cross their country the campaign was to turn out badly for Napoleon. The army occupied Lisbon but when Napoleon tried to install Joseph Bonaparte, king of Naples, as the Spanish king there was a revolt in Spain. Brianchon is said to have fought bravely in Napoleon's campaign in both Portugal and in Spain, but he was on the losing side for Napoleon's forces were defeated in both Spain and Portugal. Not only was Brianchon a brave soldier, but he was also said to be a very able one.
Brianchon remained with Napoleon's troops through the next few years but, despite a fine military career, the hard army life affected his health. In 1813, with all the fighting over, he applied to leave active service because of ill health and to take up a teaching position. It took him five years to find an appointment but eventually he was successful. In 1818 he become a professor in the Artillery School of the Royal Guard in Vincennes.
Between 1816 and 1818 while he was seeking a teaching appointment Brianchon wrote a number of papers. In these Brianchon proved several further important results in the projective study of conics. However, Brianchon did less and less research into mathematics after his teaching appointment and he turned to other interests. One paper which he did publish after taking up the appointment at Vincennes was a joint publication with Poncelet. In this paper Recherches sur la détermination d'une hyperbole équilatère, au moyen de quatres conditions donnée Ⓣ (1820) there appears a statement and proof of the nine point circle theorem. Certainly they were not the first to discover this theorem, but they were the first to give a proper proof of the theorem and also they used, for the first time, the name "nine point circle".
By 1823 Brianchon's interests were turning to teaching and to chemistry. He published on both these topics but after 1825 he gave up publishing completely and concentrated on teaching.
Article by: J J O'Connor and E F Robertson
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