MathDB

1

Part of 2008 Korea Junior Math Olympiad

Problems(1)

collinear points given product of 4 ratios =1

Source: KJMO 2008 p1

5/2/2019
In a △XYZ\triangle XYZ, points A,BA,B lie on segment ZX,C,DZX, C,D lie on segment XY,E,FXY , E, F lie on segment YZYZ. A,B,C,DA, B, C, D lie on a circle, and AZ⋅EY⋅ZB⋅YFEZ⋅CY⋅ZF⋅YD=1\frac{AZ \cdot EY \cdot ZB \cdot Y F}{EZ \cdot CY \cdot ZF \cdot Y D}= 1 . Let L=ZX∩DEL = ZX \cap DE, M=XY∩AFM = XY \cap AF, N=YZ∩BCN = Y Z \cap BC. Prove that L,M,NL,M,N are collinear.
ratiogeometryConcycliccollinear