let `P` an interioer point of a triangle `A B C` and `A P ,B P ,C P` meets the sides `B C ,C A ,A B ` in `D ,E ,F ,` respectively, Show that `(A P)/(P D)=(A F)/(F B)+(A E)/(E C)dot`

Explanation

Since A, B, C, P are co-planar, there exists four scalars `x,y,z,w` not all zero simultaneously such that
`" "xveca+yvecb+zvecc+wvecp=0`
where `" " x+y+z+w=0`
Also, `" "(xveca+wvecp)/(x+w)=(yvecb+zvecc)/(y+z)`
Hence, `" "(AP)/(PD)=-(w)/(x)-1`
Also `" "(xveca +yvecb)/(x+y)=(zvecc+wvecp)/(z+w)`
`rArr" "(AF)/(FB)=(y)/(x)`
Similarly, `" "(AE)/(EC)=(z)/(x)`
Thus, to show that `-(w)/(x)-1=(y)/(x)+(z)/(x)`
`rArr" "x+y+z+w=0` which is true.
Hence proved.