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Archive for Tháng Mười Một, 2009

simple_problem_and_its_applications

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Miquel’s pentagon theorem

linh22

Problem 5.1: Given pentagon B_1B_2B_3B_4B_5. B_1B_5\cap B_2B_3=\{A_1\}, similar for A_2,A_3,A_4,A_5(A_1B_1B_2)\cap (A_5B_1B_5)=\{C_1\}, similar for C_2,C_3,C_4,C_5. Prove that C_1,C_2,C_3,C_4,C_5 are concyclic.

Problem 5.2: See in the figure.

linh24

(A_1B_2A_2)\cap (A_1B_1B_5)=\{C_1\}, similar for C_2,C_3,C_4,C_5.  Prove that A_1C_1, A_2C_2, A_3C_3, A_4C_4, A_5C_5 are concurrent.

The solution of 5.2 can be found at here:

solution

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