Here is something I was working on a while back, it contains most of the ideas you are looking for and some others you may find useful.

In[8]:= ContourPlot[{x + 4 x^2 - 8 y + 5 x y + 4 y^2 == 0,

y == 8 x + 27, y == -x - 1, y == -(8/11) x + 25/11, y == 1/8 x ,

y == -(11/8) x + 2, y == -x + 7/13}, {x, -4, 2}, {y, -1, 5},

Epilog -> {Blue, PointSize[Large],

Point[{{-3, 3}, {-2, 1}, {-1, 3}, {0, 0}, {0, 2}, {4/3, 1/

6}, {-(8/9), -(1/9)}, {-(8/19), 49/19}, {-(17/6), 13/

3}, {-(28/9), 19/9}, {-(16/13), 23/

13}, {-(2/39) (24 + 5 Sqrt[39]),

23/13 + 10/Sqrt[39]}, {2/39 (-24 + 5 Sqrt[39]),

23/13 - 10/Sqrt[39]}}], Joined -> True, Mesh -> Full},

GridLines -> {Range[-4, 2], Range[-1, 5]}]