Use Text
. Here's an example for your case:
Show[
Plot[{4 - Sqrt[16 - (x + 3)^2],
4 + Sqrt[16 - (x + 3)^2]}, {x, -8, 3},
PlotStyle -> {Thickness[.01], Thickness[.01]}],
Graphics[{Point[{{-3, 4}, {-3, 8}, {1, 4}, {-3, 0}, {-7, 4}}],
Text[#, #, {-1, -1}] & /@ {{-3, 4}, {-3, 8}, {1, 4}, {-3, 0}, {-7, 4}} }],
ListLinePlot[{{{-3, 4}, {-7, 4}}, {{-3, 4}, {-3, 8}}, {{-3,
4}, {1, 4}}, {{-3, 4}, {-3, 0}}}]]