TE DROZ - FRNY's LNE TE FRST PURELY SYNTET PROOF ean-louis YME " " ' ' ' " bstract. We present in this article a purely synthetic proof of the Droz-Farny's line. Summary. rnold Droz-Farny. The theorem. The different proofs 4. The three lemmas 5. The synthetic proof 6. References 5. rnold Droz-Farny. rnold Droz, son of Edouard and Louise Droz, was born in La haux-de-fonds (Switzerland) on February, 856. fter his studies in the canton of Neufchatel, he went to Munich (Germany) where he attended lectures given by Felix lein, but he finally preferred geometry. n 880, he started teaching physics and mathematics in the school of Porrentruy (near asel) where he stayed until 908. e is known for having written four books between 897 and 909, two of them about geometry. ut he also published in the ournal de Mathématiques Elémentaires et Spéciales (894, 895), and in L'ntermédiaire des Mathématiciens and in the Educational Times (899) as well as in Mathesis (90). yme.-l., Forum Geometricorum (États-unis) 4 (004) 9-4 ; http://forumgeom.fau.edu/.. Gonzalez abillon, message to istoria Mathematica, ugust 8, 000, available at http://sunsite.utk.edu/math_archives/.http/hypermail/historia/aug00/0064.html. E. Ortiz, message to istoria Mathematica, ugust, 000, available at http://sunsite.utk.edu/math_archives/.http/hypermail/historia/aug00/0077.html.
s he was very sociable, he liked to be in contact with other geometers likes the talian Virginio Retali and the Spanish uan acobo Duran Loriga. n his free time, he liked to climb little mountains and to watch horse races. e married to Lina Farny who was born also in La haux-de-fonds. e died in Porrentruy on anuary 4, 9 after having suffered from a long illness.. The theorem. " " ' ' ' " f two perpendicular straight lines are drawn through the orthocenter of a triangle, they intercept a segment on each of the sides i.e. '", '", '", and the midpoints,, of these three segments are collinear [0]. Up to this day, still don't know if this theorem has been proved or not by Droz-Farny.. The different proofs. Droz-Farny's line was presented again without any proof in 995 by Ross onsberger [], after having been used by Sharygin [] in 986 as an exercise without references but he had succeeded in proving it analytically. This "remarkable theorem" as it was named by onsberger in his book has been the subject of many messages inside the yacinthos group []. f Nick Reingold [4] proposes a projective proof of it, he doesn't yet show that the considered circles intersect on the circumcircle. Darij Grinberg taking up an elegant idea of Floor van Lamoen presents a first trigonometric proof of this "rather difficult theorem" [5] which is based on the pivot theorem and applied on degenerated triangles. ut Grinberg also offers a second trigonometric proof, which starts from a generalisation of the Droz-Farny's theorem simplifying by the way the one of Nicolaos Dergiades and gives a demonstration based on the law of sines [6]. Recently, Milorad Stevanovic presents a vector proof [7]. Recently, Grinberg [7'] picks up an idea in a newsgroup on the internet and proposed a proof using inversion and a second proof using angle chasing. n this note, we present a purely synthetic proof. Finally, have discovered the proofs of. E. illyer and Sanjana resp., and also these of Droz-Farny who uses a parabola. 4. The three lemmas. Lemma (arnot). The segment of an altitude from the orthocenter to the side equals its extension from the side to the circumcircle [8]. Lemma. f a line through the orthocenter cuts the sides of a triangle at L, M, N, then (PL), (QM), (RN), the reflections of the line with regard to the sides of the triangle, are concurrent at a point F of the circumcircle [9], called the Steiner's antipoint.
Q R N M L P F Lemma or the pivot theorem. (Miquel) f a point is marked on each side of a triangle, and through each vertex of the triangle and the marked points on the adjacent sides a circle is drawn, these three circles meet at a point [0]. This result stays true in the case of tangency of lines or of two circles. Very few geometers contemporary to Miquel had realised that this result was going to become the spring of a large number of theorem. P 5. The synthetic proof. Situation : is not a right triangle
4 " " " ' ' ' " ' Let (resp., ) be the circumcircle of the triangle (resp. '", '") and ' (resp. ") be the symmetric point of in the line (resp. ). Let's point out that (resp. ) is the center of (resp. ). ccording to lemma, '' is on the circle ; '" being a diameter of the circle, ' is on the circle ; consequently, ' is the intersection of the circle, the circle, and the perpendicular to through. n the same way, we would showed that " is the intersection of the circle, the circle, and the perpendicular to through. " " M " "' ' ' ' " ' N Let "' be the symmetric of relatively to the line. ccording to lemma, ' is on the circle ; ccording to lemma applied to situated on the line ''', the lines '', "' and "'' intersect on the circle. Let N be the point of intersection of the lines '' and "'. 4
5 ccording to lemma applied to the triangle 'N' with the points ', " and situated respectively on the lines 'N, N' and '', the circles, et pass through a common point. " " M "' " 4 ' ' ' " ' N Let M be this point of concurs and 4 the circumcircle of the triangle '". Let's point out that is the center of 4. Mutatis mutandis, we would show that the circles, and 4 pass through the same point M. onclusion: the circles, and 4, all passing through and M, are coaxial. Their centers, and are collinear. Situation : is a right triangle This leads to a special case in which situation the Droz-Farny's theorem is trivial. 6. References (historical and academic) [0]. Droz-Farny., Question 4, Ed. Times 7 (899) 89-90. [] R. onsberger, Episodes in Nineteenth and Twentieth entury, M (995) 7. []. Sharygin, Problem 06, Problemas de geometria, Editions Mir (986), -. [] yacinthos@yahoogroups.com [4] N. Reingold, On the Droz-Farny theorem, yacinthos messages #78 du /08/00, # 784. [5] D. Grinberg, On the Droz-Farny theorem, yacinthos messages 68, 64, 645, 0-//00 F. v. Lamoen, On the Droz-Farny theorem, yacinthos messages 640, 644, //00.-P. Ehrmann, On the Droz-Farny theorem, yacinthos messages 650, 657, //00. [6] D. Grinberg, From the omplete Quadrilateral to the Droz-Farny Theorem, avaible from http://de.geocities.com/darij_grinberg. [7] M. Stevanovic, Droz-Farny theorem again, Message yacinthos 90, 5/0/004. [7'] D. Grinberg, synthetic proof of the Droz-Farny theorem, yacinthos message 9845, 0/06/004. D. Grinberg, The Droz-Farny theorem is a piece of cake, yacinthos messages 0/06/004. [8] L. N. M. arnot, De la corrélation des figures géométriques (80), n 4, p.0. R.. ohnson, dvanced Euclidean Geometry, Dover (960), n 54, p.6. [9] T. Lalesco, La Géométrie du Triangle (96), Rééditions acques Gabay, Paris (987), n -8, p.99. 5
6 R.. ohnson, dvanced Euclidean Geometry (95) n, Dover (960), n. [0]. Miquel, Mémoire de Géométrie, ournal de mathématiques pures et appliquées de Liouville vol., (88) 485-487. R.. ohnson, dvanced Euclidean Geometry (95) Dover (960), n 84, p.. 6