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3 straight lines intersecting n circles, equal segments wanted and given

Source: 1997 Romania NMO IX p1

August 13, 2024
geometrycirclesequal segments

Problem Statement

Let C1,C2,...,CnC_1,C_2,..., C_n , (n≥3)(n\ge 3) be circles having a common point MM. Three straight lines passing through MM intersect again the circles in A1,A2,...,AnA_1, A_2,..., A_n ; B1,B2,...,BnB_1,B_2,..., B_n and X1,X2,...,XnX_1,X_2,..., X_n respectively. Prove that if A1A2=A2A3=...=An−1AnA_1A_2 =A_2A_3 =...=A_{n-1}A_n and B1B2=B2B3=...=Bn−1BnB_1B_2 =B_2B_3 =...=B_{n-1}B_n then X1X2=X2X3=...=Xn−1Xn.X_1X_2 =X_2X_3 =...=X_{n-1}X_n.
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