For three vectors to be coplanar we have
Which gives
Substituting equation 2 in 1 we have
So we have
= (x - 2) (y - 2) ( z - 2) - (x - 2) (y -1) (z -1) - (x - 1) (y - 2) (z - 1) - (x - 1) (y - 1) (z - 2) + 2(x - 1)(y - 1)(z - 1) = 0
Solving we have x + y + z = 4
Hence cosec2A + cosec2B + cosec2C = 4