http://www.cpohoata.com Website for Classical Euclidean Geometry Fri, 07 Aug 2009 09:11:55 +0000 http://backend.userland.com/rss092 en Cosmin Pohoata, A note on the anticomplements of the Fermat points, Forum Geometricorum 9 (2009) 119-123. (Link to the paper's abstract) We show that each of the anticomplements of the Fermat points is common to a triad of circles involving the triangle of reflection. We also generate two new triangle centers ... http://www.cpohoata.com/2009/06/18/657/ Andrei Frimu, Yimin Ge, Daniel Kohen, David Kotik, Hojoo Lee, Soo-Hong Lee, Cosmin Pohoata, Ho Chung Siu, Peter Vandendriessche, Ofir Gorodetsk, Alexander Remorov, Darij Grinberg, Harun Siljak, Marin Misur, Problems in Elementary Number Theory 2 (Spring 2009) (43 pages; link to the PDF file). In this volume, you will meet the ... http://www.cpohoata.com/2009/06/18/pen/ Cosmin Pohoata, Vladimir Zajic, Generalized Apollonian circles, submitted. (Link to the PDF file) The three Apollonius circles of a triangle, each passing through a triangle vertex, the corresponding vertex of the cevian triangle of the incenter and the corresponding vertex of the circumcevian triangle of the symmedian point, are coaxal. Similarly ... http://www.cpohoata.com/2008/12/27/633/ Andrei Frimu, Yimin Ge, Ofir Gorodetsky, Daniel Kohen, David Kotik, Hojoo Lee, Soo-Hong Lee, Cosmin Pohoata, Ho Chung Siu, Peter Vandendriessche, Problems in Elementary Number Theory 1 (Fall 2008) (39 pages; link to the PDF file). In this volume, you will meet the solutions of the problems from the first season ... http://www.cpohoata.com/2008/10/25/problems-in-elementary-number-theory-1-fall-2008/ Hojoo Lee, Tom Lovering, Cosmin Pohoata, INFINITY, 2008. (204 pages; both TeX and PDF are available for download) In this never-ending project, which bears the name Infinity, we offer a delightful playground for young mathematicians and try to continue the beautiful spirit of IMO and MathLinks. Infinity begins with a chapter ... http://www.cpohoata.com/2008/10/17/infinity/ Cosmin Pohoata, A combinatorial proof of Lindstr"om's theorem on unions of subsets of a finite set. (Link to the PDF file) In this note, we give a combinatorial proof of Bernt Lindstr\"om's theorem that if A_{1}, A_{2},..., A_{n+1} are nonempty subsets of an n-element set, then we can find two ... http://www.cpohoata.com/2008/10/02/440/ Cosmin Pohoata, Geometric constructions of the mixtilinear incircles, Crux Mathematicorum with Mathematical Mayhem, to appear, 2008. (Link to the submitted PDF file) The term mixtilinear incircles of a triangle was introduced by L. Bankoff [1] naming the three circles each tangent to two sides and to the circumcircle internally.  In the ... http://www.cpohoata.com/2008/09/24/geometric-constructions-of-the-mixtilinear-incircles/ Cosmin Pohoata, A new method of trisection, Gazeta Matematica, 7-8 (2008), 350-351. (in Romanian) - an English version will be attached In this short note, we present D. A. Brooks' trisection method (of a given angle) from an elementary point of view. http://www.cpohoata.com/2008/09/24/on-a-new-method-of-trisection/ Cosmin Pohoata, On a theorem regarding lattice pentagons, Mathematical Reflections, 5 (2008). (Link to the journal’s PDF file) My original variant can be downloaded from here. In this note, we give a short proof of a Russian refinement of Arkinstall's lattice pentagon theorem. http://www.cpohoata.com/2008/09/20/on-a-theorem-regarding-lattice-pentagons/ Gabriel Dospinescu, Mircea Lascu, Cosmin Pohoata, Marian Tetiva, An elementary proof of Blundon's Inequality, Journal of Inequalities in Pure and Applied Mathematics, Volume 9, Issue 4, Article 100, 2008. (Link to the official PDF file) In this note, we give an elementary proof of Blundon's Inequality. We make use of a ... http://www.cpohoata.com/2008/08/08/an-elementary-proof-of-blundons-inequality-2/ Cosmin Pohoata, The Euler reflection point, The Harvard College Mathematics Review, 2009, to appear. (Link to the PDF file) The Euler reflection point of a triangle is known in literature as the common point of the reflections of its Euler line OH into each of its sidelines, where O, and H ... http://www.cpohoata.com/2008/07/23/the-euler-reflection-point1/ Cosmin Pohoata, Son Hong Ta, A short proof of Lamoen's generalization of the Droz-Farny line theorem, preprint. (Link to the PDF file) In 1899, Arnold Droz-Farny discovered the following beautiful result, known nowadays as the Droz-Farny line theorem: If two perpendicular straight lines are drawn through the orthocenter of a triangle, ... http://www.cpohoata.com/2008/07/23/a-short-proof-of-lamoens-generalization-of-the-droz-farny-line-theorem/ Cosmin Pohoata, Boole's formula as a consequence of Lagrange's interpolating polynomial theorem, INTEGERS: The Electronic Journal of Combinatorial Number Theory, 8 (2008), #A23. (Link to the PDF file). Another variant of the paper can be downloaded from http://arxiv.org/abs/0807.1133.) We present a slightly more general version of Boole's additive formula for factorials as ... http://www.cpohoata.com/2008/07/23/booles-formula-as-a-consequence-of-lagrages-interpolating-polynomial-theorem/ Cosmin Pohoata, A short proof of Lemoine's theorem, Forum Geometricorum 8 (2008), 97-98. (Link to the paper's abstract) We give a short proof of Lemoine's theorem that the Lemoine point of a triangle is the unique point which is the centroid of its own pedal triangle. We make use of ... http://www.cpohoata.com/2008/07/23/a-short-proof-of-lemoines-theorem/ Cosmin Pohoata, From Neuberg-Pedoe back to Hadwiger-Finsler. (Link to the PDF file) In this note, we give a new parametrized version of the Hadwiger-Finsler Inequality. Within our proof, we make use of a preliminary result, which can be interpreted as a corollary of the Neuberg-Pedoe Inequality. http://www.cpohoata.com/2008/07/23/an-elementary-proof-of-blundons-inequality/ Cezar Lupu, Cosmin Pohoata, Sharpening Hadwiger-Finsler's inequality, Crux Mathematicorum with Mathematical Mayhem, 2 (2008), 97-101. (Link to the extracted PDF file); based on this article, in 2008, the authors were awarded with the Traian Lalescu Medal for their undergraduate research activity (though technically I am not an undergraduate yet). The Hadwiger-Finsler ... http://www.cpohoata.com/2008/07/23/sharpening-hadwiger-finslers-inequality/ Cosmin Pohoata, On the Parry reflection point, Forum Geometricorum 8 (2008), 65-70. (Link to the paper's abstract) We give a synthetic proof of C. F. Parry's theorem that the  reflections in the sidelines of a triangle of three parallel lines through the vertices are concurrent if and ... http://www.cpohoata.com/2008/07/23/on-the-parry-reflection-point/ Cosmin Pohoata, On a remarkable concurrency, Gazeta Matematica, 2 (2008), 65-70. (in Romanian) - an English version will be attached We give five different proofs of a concurrency due to Jean-Pierre Ehrmann (see Hyacinthos message #6966): Let ABC be a triangle and let D, E, F be the tangency points of ... http://www.cpohoata.com/2008/07/23/on-a-remarkable-concurrency/ Cosmin Pohoata, Paul Yiu, On a product of two points induced by their cevian triangles, Forum Geometricorum, 7 (2007), 169-180. (Link to the paper's abstract) The intersections of the corresponding sidelines of the cevian triangles of two points P_0 and P_1form the anticevian triangle of a point T(P_0,  P_1). We prove ... http://www.cpohoata.com/2008/07/23/on-a-product-of-two-points-induced-by-their-cevian-triangles/ Cosmin Pohoata, Harmonic Division and Its Applications, Mathematical Reflections, 4 (2007). (Link to the journal's PDF file) Let A, B, C, D be four points lying in this order on a given line. The quadruple (A, B, C, D) is called a harmonic division if and only if latex BA/BC=-DA/DC, or ... http://www.cpohoata.com/2008/07/22/on-the-gigi-point/ Cezar Lupu, Cosmin Pohoata, About a nice inequality, Mathematical Reflections, 1 (2007). (Link to the journal's PDF file) We consider two proofs for Darij Grinberg's a^2+b^2+c^2 + 2abc +1 \geq 2(ab+bc+ca), which lead us to solving several other three-variable inequalities: one from the Romanian National Mathematics Olympiad from 2004, one from ... http://www.cpohoata.com/2008/07/21/about-a-nice-inequality/ Cosmin Pohoata, A possibly new proof of Euler's Triangle Inequality. (Link to the PDF file) In this note, we give a possibly new proof of Euler's Inequality that in any triangle with circumradius R and inradius r, we have that R \geq 2r. http://www.cpohoata.com/2008/07/19/a-possibly-new-proof-of-eulers-triangle-inequality/ Cosmin Pohoata, Remarks on the Japanese Theorem, Arhimede Magazine, 1 (2007), 6-9. (in Romanian) - an English version will be attached; this article was awarded with the First Prize and Gold Medal at the International Arhimede Symposium in Pure and Applied Mathematics, held in Bucharest, 2008. The Japanese theorem states that ... http://www.cpohoata.com/2008/07/15/290/