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If D ,Ea n dF are three points on the si...

If `D ,Ea n dF` are three points on the sides `B C ,C Aa n dA B ,` respectively, of a triangle `A B C` such that the `(B D)/(C D),(C E)/(A E),(A F)/(B F)=-1`

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`xveca+ yvecb+zvecc+ tvech =0` such that
`x+y+z+t=0" "` (i)
`x veca + yvecb= -(zvecc+ tvech)`
and `x+y= - (z+t)`
`therefore (xveca+yvecb)/(x+y) = (zvecc+tvech)/(z+t)" "`(ii)
Position vector of `F= (xveca+yvecb)/(x+y)`

Hence, F divides AB in the ratio y/x.
`(AF)/(FB) = (y)/(x)`
Similarly, `(BD)/(CD)= (z)/(y) and (CE)/(AE)= (x)/(z)`
`rArr (AF)/(FB) *( BD)/(CD)*(CE)/(AE) = -1 `
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