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Ratio of lengths being integers.

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January 28, 2011
ratiogeometrygeometry unsolved

Problem Statement

For a point OO inside a triangle ABCABC, denote by A1,B1,C1,A_1,B_1, C_1, the respective intersection points of AO,BO,COAO, BO, CO with the corresponding sides. Let n1=AOA1O,n2=BOB1O,n3=COC1O.n_1 =\frac{AO}{A_1O}, n_2 = \frac{BO}{B_1O}, n_3 = \frac{CO}{C_1O}. What possible values of n1,n2,n3n_1, n_2, n_3 can all be positive integers?
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