Points `A( vec a),B( vec b),C( vec c)a n dD( vec d)` are relates as `x vec a+y vec b+z vec c+w vec d=0` and `x+y+z+w=0,w h e r ex ,y ,z ,a n dw` are scalars (sum of any two of `x ,y ,zn a dw` is not zero). Prove that if `A ,B ,Ca n dD` are concylic, then `|x y|| vec a- vec b|^2=|w z|| vec c- vec d|^2dot`

Explanation

From the given conditions, it is clear that points `A(veca), B(vecb), C(vecc) and D(vecd)` are coplanar.
Now, `A, B, C and D` are concyclic. Therefore,
`" "APxxBP = CPxxDP`
`" "|(y)/(x+y)||veca-vecb||(x)/(x+y)||veca-vecb|=|(w)/(w+z)||vecc-vecd||(z)/(w+z)||vecc-vecd|`
`" "|xy||veca-vecb|^(2)= |wz| |vecc-vecd|^(2)`