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in Mathematics by (53.3k points)

Suppose a straight line meets the sides BC, CA and AB of a triangle at points D, E and F, respectively. Then show that BD/DC. CE/EA.AF/FB

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Let the coordinates of A, B and C be (x1, y1), (x2, y2) and (x3, y3). See Fig.. Suppose L is the line whose equation is L = ax + by + c = 0 meeting the sides in D, E and F. Therefore,  we get

Hence

BD/DC.CE/EA.AF/FB

Note: Either the line l meets all three sides externally or two sides internally and one side externally. That is why product of the ratios is equal to −1.

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