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In the plane of ΔABC, let O be a point (not on any side). If the lines AO, BO and CO meet the opposite sides BC, CA and AB at points D, E and F, respectively, then show that

BD/DC.CE/EA.AF/FB = +1

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Without loss of generality, we may assume that O is the origin and points A, B and C are (x1, y1), (x2, y2) and (x3, y3), respectively. See Fig.. Observe that either all the three lines (bar)AO  , (bar)BO  and (bar)CO divide the sides BC, CA and AB internally or two of them divide two sides externally and one divides the third side internally. Now, the equations of the lines (bar)AO  , (bar)BO and (bar)CO are, respectively, xy− x1y = 0, xy− x2y = 0 and xy3 − x3y = 0. Therefore, by , we get

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