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2 A DA M BESENYEI,Figure 1 Brocard points, and equality can hold if only if the triangle is equilateral. It seems lesser known that a generalization of the inequality 2 was for. mulated in the paper 3 from 1945 by the Russian mathematician and physi. cist Nikolai Aleksandrovich Dmitriev 1924 2000 and mathematician Eu. gene Borisovich Dynkin 1924 They stated the following nice geometrical. result as a lemma in their paper we keep the term lemma. Lemma Let P be an arbitrary point in the interior of a convex n gon. A1 A2 An and denote An 1 A1 see Fig 2 Then,min P Ak Ak 1. Equality occurs if and only if A1 A2 An is a regular n gon. In fact the case of equality was not explicitly mentioned by Dmitriev and. Dynkin but it follows from their argument We first present their proof by. giving some details on a minor step that they left to the reader Then we. recall some related problems from the past few decades and solve them with. the lemma of Dmitriev and Dynkin,Proof of the Lemma. We prove the lemma by contradiction Assume that P Ak Ak 1 for. every k 1 n where we use the notation 2 n for brevity. Figure 2 Minimal angle lemma,BROCARD ANGLE AND GEOMETRICAL GEM 3. Figure 3 Proof of the Lemma, Since P Ak Ak 1 there is a point Nk on the segment P Ak 1 such that.
P Ak Nk see Fig 3 Denote by k the angle P Nk Ak Then the sine. theorem in the triangle P Ak Nk implies,sin k P Ak P Ak. sin P Nk P Ak 1, The product of the above inequality for k 1 n yields. sin 1 sin 2 sin n P A1 P A2 P An,sin sin sin P A2 P A3 P A1. 3 sin 1 sin n sin n,On the other hand,k Ak P Ak 1,n Ak P Ak 1 n 2 n. But the product sin 1 sin n in which the angles k satisfy. 0 1 0 n k n,attains its maximum for 1 n So,sin 1 sin n sin n.
which contradicts 3 The value and location of the maximum of sin 1. sin n which was not justified by Dmitriev and Dynkin follows easily from. the inequality of arithmetic and geometric means combined with Jensen s. inequality for the concave sine function on the interval 0 as they might. also had the same reasoning in their minds Indeed,sin 1 sin n n. sin 1 sin n,4 A DA M BESENYEI, Observe that the above proof yields also the case of equality in the Lemma. In case of equality P Ak Ak 1 for every k and Nk Ak 1 is also. possible but still 0 k Then in inequality 3 equality is also. possible and in fact equality is the only possibility as we showed Thus. 1 n which means k for every k therefore A1 A2 An is a. regular n gon,Many birds with one stone, We now give a collection of problems from the past few decades to illus. trate the variety of ways in which the above lemma has appeared without. the names of Dmitriev and Dynkin being mentioned which might be due. to the inaccessibility of their paper, Brocard angle For n 3 the Lemma implies that the Brocard angle. of a triangle is at most 6 and equality holds provided that the triangle. is equilateral Moreover if a convex n gon admits a Brocard point i e a. point P such that the angles P Ak Ak 1 are equal then the corresponding. Brocard angle is at most 2 n with equality in the case of a regular. n gon We note that there exist n gons that do not have Brocard points. see 1 for such an example, Contest problems The Lemma for the particular case n 3 was a prob.
lem of the International Mathematical Olympiad in 1991 see 11 p 23. This book contains three different solutions to the problem one of them. based on the maximal value of the Brocard angle in a triangle likely to. be well known for students preparing for Olympiads The approach of. applying the inequality of arithmetic and geometric means and Jensen s in. equality comes also from there This idea appears also in the research report. 5 where the author provides an elegant proof of the maximal value of the. Brocard angle in a way very similar to the above presented applied to the. particular case n 3, For n 4 the Lemma appeared as a problem of a national contest in. India in 1991 see 2 p 55 It is interesting that the two solutions given. therein are rather algebraic and trigonometric, Monthly problems The case of equality in the Lemma appeared two. times in the Problem Section of the American Mathematical Monthly in the. years 2000 2001 Problem 10824 see 10 asked to show that if there is a. point P in a triangle such that the angles P AB P BC P CA 30. then the triangle is equilateral Problem 10904 claimed analogous statement. for n 4 see 6 moreover it also raised the question of possible general. izations to n gons This latter subproblem was marked with an meaning. that no solutions were then available but as we see there did already exist. a published solution The Monthly published a solution due to A Nijenhuis. who formulated essentially the lemma of Dmitriev and Dynkin with a proof. very similar to theirs he used the method of Lagrange s multipliers to find. the maximum of a product of trigonometric functions. BROCARD ANGLE AND GEOMETRICAL GEM 5,Historical remarks. The aim of the 1945 paper of Dmitriev and Dynkin was to determine. the location of eigenvalues of stochastic matrices in the complex plane i e. matrices with nonnegative entries and with each row summing to 1 This. question was initiated by Andrey Kolmogorov 1903 1987 in his seminar. on Markov chains at Moscow State University The problem was partially. answered independently by Dmitriev and Dynkin who were students that. time at the university They published two papers on the subject in the. years 1945 46 in which they transformed the linear algebra question to a. geometrical one Their first paper contained the Lemma of the present. paper which is a nice example of a bridge between two distant concepts of. mathematics, It is remarkable how brilliantly talented the two young men were Dmitriev. enrolled the university at the age of 14 Dynkin at the age of 16 Later both. became prominent scientists The book 9 p 29 recalls a funny episode. when Kolmogorov was asked for advice on the matter of use computers in. the physics institue Kolmogorov replied Why do you need those comput. ers You have Kolya Nikolai Dmitriev don t you We refer to 12 for. a brief biography of Dmitriev and to 4 a lecture recorded at Cornell Uni. versity where Professor Dynkin recounts his seventy years in mathematics. including the above mentioned problem, Acknowledgment The author wishes to thank Pe ter L Simon for men.
tioning the geometrical lemma and Craig Smoryn ski for sending a copy of. the paper 3,References, 1 A Ben Israel S Foldes Complementary halfspaces and trigonometric Ceva Brocard. inequalities for polygons Math Inequal Appl 2 1999 307 316. 2 T Andreescu Z Feng eds Mathematical Olympiads 1998 1999 Problems and So. lutions from Around the World MAA Washington D C 2000. 3 N Dmitriev E Dynkin On the characteristic roots of stochastic matrices C R. Doklady Acad Sci URSS 49 1945 159 162, 4 E Dynkin Seventy Years in Mathematics Lecture recorded at Cornell University. in 2010 Eugene B Dynkin Collection of Mathematics Interviews Cornell University. Library http dynkincollection library cornell edu, 5 S Foldes Another proof of the Brocard angle limit theorem RUTCOR Research. Report RRR 9 98 February 1998, 6 O Furdui A Nijenhuis A square property Solution to problem 10904 Amer Math. Monthly 109 2002 860 861, 7 R Honsberger Episodes in the Nineteenth and Twentieth Century Euclidean Geom.
etry MAA Washington D C 1995, 8 R A Johnson Advanced Modern Geometry An Elementary Treatise on the Geom. etry of the Triangle and the Circle Dover Publ Inc New York 1960. 9 P N Lebedev Physics Institute eds Andrei Sakharov Facets of a Life E ditions. Frontie res Gilf sur Yvette France 1991, 10 H Lee M A P Bernstein Brocard angle 30 degrees Solution to Problem 10824. Amer Math Monthly 109 2002 481 482, 11 I Reiman International Mathematical Olympiads Vol III 1991 2004 Anthem. Press London New York 2005, 12 V S Vladimirov et al Nikolai Aleksandrovich Dmitriev obituary Russ Math Surv. 56 2001 403 406,6 A DA M BESENYEI,A da m Besenyei, Department of Applied Analysis Eo tvo s Lora nd University H 1117 Budapest.
Pa zma ny P se ta ny 1 C Hungary,E mail address badam cs elte hu. THE BROCARD ANGLE AND A GEOMETRICAL GEM FROM DMITRIEV AND DYNKIN AD AM BESENYEI Abstract In a celebrated paper on the eigenvalues of special matrices N Dmitriev and E Dynkin formulated and proved a nice geometrical