tailieunhanh - Tuyển tập các bài hình học vô địch thế giới P4

Tuyển tập các bài hình học vô địch thế giới P4 là tài liệu tham khảo, nó đặc biệt hay và thú vị, các bạn nào đam mê toán có thể nghiên cứu, tài liệu tuyển tập các bài hình học từ khó đến cực khó trên thể giới, được viết bằng tiếng anh, Chúc các bạn ôn thi tốt. | SOLUTIONS 301 similarly prove that X belongs to the circles of nine points of triangles BCD CDA and DAB. ---- . Let Xi be the projection of X on l. Vector aAAi ßBBi 7CCi is the projection of vector aAX 1 ßBX 1 7CX 1 to a line perpendicular to l. Since aA i ßB i 7C i aA ßB 7C a ß 7 X 1 _ and ahi ßB 7C 0 by Problem we get the statement required. . T . ---1 ---1 -------- 1 . I r. T . 1 1 . Let a A1A2 A3 A4 A2n-1A2n and a 0. Introduce the coordinate - system directing the Ox-axis along vector a. Since the sum of projections of vectors A1A2 A3A4 . A2n_1A2n on Oy is zero it follows that the length of a is equal to the absolute value of the difference between the sum of the lengths of positive projections of these vectors to the Ox-axis and the sum of lengths of their negative projections. Therefore the length of a does not exceed either the sum of the lengths of the positive projections or the sum of the lengths of the negative projections. It is easy to verify that the sum of the lengths of positive projections as well as the sum of the lengths of negative projections of the given vectors on any axis does not exceed the diameter of the circle . does not exceed 2. . In the proof of the equality of vectors it suffices to verify the equality of their projections minding the sign on lines BC CA and AB. Let us carry out the proof for example for the projections on line BC where the direction of ray BC will be assumed to be the positive one. Let P be the projection of point A on line BC and N the midpoint of BC. Then d A b2 a2 - c2 a b2 - c2 PN PC CN 2a 2 20- PC is found from the equation AB2 BP2 AC2 CP2 . Since NM NA 1 3 _ _ 1 __ b _r2 the projection of MO on line BC is equal to 1PN _a . It remains to notice that the projection of vector a3na b3nb c3nc on BC is equal to b3 sin y c3 sin ß b3c c3b 2R abc b2 c2 2pb2 c2 2R a a . Let the inscribed circle be tangent to sides AB BC and CA at points U V and - -- - - - -- W respectively. We have to prove