We have the points A(2,-1),B(-1,1),C(1,3). Find the coordinates of D point, knowing that ABCD parallelogram.

ABCD is a parallelogram. The coordinates of D is not known. Let us find it.

Since in the parallelogram AB||CD, slope of AB=slope of DC.

Slope of AB=(BY2-Ay1)/(Bx2-Ax1)=(1-(-1))/(-1-2)=2/(-3)

Slope of DC = (y-3)/(x-1) . Equating the two slopes we get(y-3/(x-1) =-2/3=>y-3=(-2/3)(x-1)--------(1)

Similarly, equating the slopes of the other two parallel sidesAD and BC we get  (y+1)/(x-2)=(3-1)/(1+1)=>y+1=(x-2)---(2)

From the equations (1) and (2) , by subraction:

-4=(-2/3-1)x+2/3+2=>  (5/3)x=6+2/3=20/3=>

x=20/5=4. Substituting in (2) , y+1=4-2=2 or y=1

Therefore, the coordinates of D is(4,1).

Check:

You can verify whether AB=DC and AD=BC:

AB^2=(-2-1)^2+(1+1)^2=13, CD^2=(4-1)^2+(1-2)^2=13.cheks OK.Check yourself the other two sides.