The necessary and sufficient conditions for the three-point m, n and C*** lines are,

The existence of real numbers m, n∈R and m+n= 1

OC=m*OM+n*ON

1) If three points m, c, N***

Then MC=k*MN

∴OC-OM=k*(ON-OM)

That is OC=OM-k*OM+k*ON.

=( 1-k)*OM+k*ON

Let 1-k = m and k = n, then m+n= 1.

∴ There are real numbers m, n∈R and m+n= 1, so that

OC=m*OM+n*ON

2) If there is a real number m, n∈R and m+n= 1

OC=m*OM+n*ON

therefore

OC=m*OM+( 1-m)*ON

=m*OM-m*ON+OM

∴OC-OM=m(OM-ON)

∴MC=m*NM

∴M,C,N three-point * * * line

Obtain a certificate

According to the above theorem, in this problem,

The three-point * * * lines of M, C and N are

OC=2xOM+2yN。

∴2x+2y= 1

That is, the point (x, y) is on the straight line 2x+2y= 1.

The answers are all right.