ࡱ> oqn`".bjbjss R#i  &/&/&/8^/1T Z|r1 <:<<<DADADAZ!Z!Z!Z!Z!Z!Z$\h^EZ }BA@@A}B}BEZ<<FZZIII}Bf< <ZI}BZIIcWht  X<f1 B&/eCjWYTpZ0ZWd_Fd_ Xd_ X,DA>A,IA$ADADADAEZEZH^DADADAZ}B}B}B}B d-$ -   THE BOOK OF GEOMETRICAL INEQUALITIES AS A COLLECTION OF THE CREATIVE WORK OF THE STUDENTS NAIRI M.SEDRAKYAN The second enlarged edition of the book Geometrical inequalities (404 pages) was published in Russian in 2006. It contains more than 1000 problems, a significant part of which include are the result of the creative work of the pupils (these is noted in the book), the other part can be used for stimulating creativity of the students. For promoting the creativity of the students it is very important, that after considering the special cases they would be able to generalize. Also they should be able to find important applications of the general facts. The problems 13.31-13.33 of the named book can be a good example of acquiring this qualities by students and the report will be devoted to the discussion of these problems. In 1997 I proposed for the Armenian Olympiad the following problem Example 1. For arbitrary points  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  prove the inequality:  EMBED Equation.DSMT4 . That inequality can be generalized to the following: Example 2. For any hexagon  EMBED Equation.DSMT4  prove the inequality  EMBED Equation.DSMT4 . That inequality was the source of a new inequality, which in 2006 was presented on the second Zhautikow Olympiad on mathematics and physics, January 15-20, Almaty, Kazakhstan. Example 3. In the convex hexagon  EMBED Equation.DSMT4  it is known that  EMBED Equation.DSMT4 . Prove that  EMBED Equation.DSMT4 . Now well give two solutions of the last inequality. We shall prove that for any hexagon  EMBED Equation.DSMT4  the inequality  EMBED Equation.DSMT4  is true, the equality taking place if and only if  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Indeed, let  EMBED Equation.DSMT4 . Thus we have that  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . (1) Consequently we find that  EMBED Equation.DSMT4  hence  EMBED Equation.DSMT4 . (2) Note that the equality in (2) holds if and only if it holds in the inequalities of (1) and  EMBED Equation.DSMT4 , i.e.  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . From the conditions of the problem we find that  EMBED Equation.DSMT4 , consequently,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Consider now the parallelogram  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4  is also a parallelogram, consequently,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 , thus  EMBED Equation.DSMT4 , hence  EMBED Equation.DSMT4 , or  EMBED Equation.DSMT4 . Second solution. Let  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Since  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 . Similarly, we find, that  EMBED Equation.DSMT4 , hence one can construct a triangle with sides  EMBED Equation.DSMT4 . Note, that  EMBED Equation.DSMT4  (see Fig.1, Fig.2). Indeed, if  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4  (1)  Consider a parallelogram  EMBED Equation.DSMT4  (Fig.1). We have  EMBED Equation.DSMT4 , hence  EMBED Equation.DSMT4 . On the other hand  EMBED Equation.DSMT4 , which is impossible. Similarly, we find that  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , but since  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . We have found that  EMBED Equation.DSMT4 , consequently  EMBED Equation.DSMT4 , which means that  EMBED Equation.DSMT4 . Since  EMBED Equation.DSMT4 , we have  EMBED Equation.DSMT4  (see Fig.1). Similarly, we find that  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , then the quadrangles  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  are parallelograms, and since  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4 . The Example 3 can be generalized to Example 4. For a convex 2n-agon  EMBED Equation.DSMT4  prove the inequality  EMBED Equation.DSMT4   EMBED Equation.DSMT4 . The next inequality is closely connected to the inequality of the next example. Example 5. For a convex 2n-angler  EMBED Equation.DSMT4  prove the inequality  EMBED Equation.DSMT4   EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4  are midpoints of the segments  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , correspondingly. It is interesting, that for  EMBED Equation.DSMT4  the solution of the Example 5 brings the solution of the problem of the IMO2003. Example 6. Given a convex hexagon, for each pair of opposite sides of which the following condition is satisfied: the ration of the distance between the midpoints of these sides to the sum of the lengths of these sides is equal to  EMBED Equation.DSMT4 . Prove that all angles of the hexagon are equal. (Convex hexagon  EMBED Equation.DSMT4  has three pairs of opposite sides:  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 ). It turns out, that the solution of the Example 6 is connected with the generalization of the Erdsh-Mordell theorem. Example 7. Let  EMBED Equation.DSMT4  be a point inside the convex  EMBED Equation.DSMT4 -gon  EMBED Equation.DSMT4 . Denote  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4 . Prove that  EMBED Equation.DSMT4 . Now well present solutions of some of the above examples, which I have sent to Titu Andreescu and the American Monthly in 2005. Solution. Lets denote  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . We have that  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 , hence  EMBED Equation.DSMT4  since  EMBED Equation.DSMT4 . Consequently,  EMBED Equation.DSMT4  (1) Its clear that  EMBED Equation.DSMT4 , and if we shall prove that  EMBED Equation.DSMT4 (2) then from (1) and (2) we find that  EMBED Equation.DSMT4 . Consider on the coordinate plane the points  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  (see Fig.3), such that  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 .  According to the law of cosines  EMBED Equation.DSMT4 , consequently, the inequality (2) takes the form:  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  (3) If we now shall prove that for any numbers  EMBED Equation.DSMT4  (4) then we can use the inequality (4) for  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , we shall prove the inequality (3). Thus we have to prove that for  EMBED Equation.DSMT4  and any numbers  EMBED Equation.DSMT4  holds the inequality  EMBED Equation.DSMT4 . (5) Its not difficult to check that  EMBED Equation.DSMT4  (6) Consequently the inequality (5) is true. To prove (6) it remains to note that  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Example 4. Solution. Let  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Note that  EMBED Equation.DSMT4  We have that  EMBED Equation.DSMT4 , consequently,  EMBED Equation.DSMT4 , Hence  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4 . Similarly. we find that  EMBED Equation.DSMT4 ,.,  EMBED Equation.DSMT4 . Since  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 , we have  EMBED Equation.DSMT4 . Summing these inequalities we get  EMBED Equation.DSMT4   EMBED Equation.DSMT4  (1) Also we have  EMBED Equation.DSMT4  (2) (see the solution of the example 7), hence from (1) and (2) we find that  EMBED Equation.DSMT4   EMBED Equation.DSMT4 , or  EMBED Equation.DSMT4   EMBED Equation.DSMT4 . Remark. The obtained estimates are exact, since for the regular 2n-gon we have equality. Example 5. Solution. We have that  EMBED Equation.DSMT4 , consequently,  EMBED Equation.DSMT4 . Similarly we find  EMBED Equation.DSMT4 ,.,  EMBED Equation.DSMT4 . We have  EMBED Equation.DSMT4 , consequently,  EMBED Equation.DSMT4 . Summing the obtained inequalities we have  EMBED Equation.DSMT4   EMBED Equation.DSMT4 . According to the solution of the Example 4 we get  EMBED Equation.DSMT4   EMBED Equation.DSMT4 . Hence  EMBED Equation.DSMT4  EMBED Equation.DSMT4 . ABOUT THE AUTHOR Nairi M.Sedrakyan Armenia E-mail:  HYPERLINK "mailto:gagri@yerphi.am" gagri@yerphi.am     Nairi M. 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