radius = 20;
circ = RegionPlot[
x^2 + y^2 <= radius^2, {x, -radius, radius}, {y, -radius, radius},
Mesh -> 20, MeshStyle -> {{Red}, {Blue}}, BoundaryStyle -> Black,
PlotStyle -> None]
circ2 = Show[circ, Graphics[{PointSize[0.025], Point[{10, 10}]}]]

You could have used ListPlot
too, but the point here is that Show
combined Graphics
into a new Graphics
, so you need to feed it with Graphics (ListPlot
returns a Graphics
object)
circ3 = Show[circ,
ListPlot[{{10, 10}}, PlotStyle -> PointSize[0.025]]]
Another option perhaps more similar to what you were trying could be
Show[circ, Epilog -> {PointSize -> Large, Point[{{10, 10}}]}]
Graphics
are composed of primitives, directives and options.
Primitives are the shapes (Line
, Point
, Circle
, etc).
Directives are stuff that affect the shapes, such as color settings, point size, line type, arrow heads, etc. In the same Graphic
object you can combine several directives, affecting different primitives (a red circle with a black dot)
Options are general to the graphic. AspectRatio
, stuff about the axes, etc..
So, a Graphics
object is Graphics[{list of primitives and directives}, options]
What Show
does is take several Graphics
, combine its primitives and directives, and optionally change the general options.
PlotStyle
is an option for functions that automatically create Graphics
, like Plot
and his family. It is NOT a Graphics
option. In those functions, it allows among other things to set directives to the stuff you're plotting. So if you're used to writing Plot[f[t], {t, 0, 10}, PlotStyle->something, otheroptions->othervalues]
then that something is probably a directive. And probably, otheroptions are general graphic options (or other particular options of the plotting function such as Filling
, etc).
So, let's see what you were trying to do with your code
Show[circ,
Epilog -> Inset[ListPlot[{10, 10}], PlotStyle -> PointSize[0.025]] ]
You were trying to create a new Graphics
with the same primitives and directives than circ, but with an extra option Epilog
. Furthermore, you were trying to add as options to the Graphics
something that is not an option :P.
Epilog
just adds more primitives to the Graphics
object after it is rendered. It is more or less the same as adding more primitives to the list, except that it allows to do it as an option so it's convenient to add annotations in functions such as Plot
. The fact that they are added after the Graphics
was rendered also means you can mix 2D and 3D primitives. No different in your case than writing the following
Show[circ, Graphics[{Inset[ListPlot[{10, 10}]]}]]
Moving on. Inset
takes a Graphics
, Image
, cell expression, string, etc, and turns it into a primitive. So, the Graphics
object generated by ListPlot
now is taken as a whole, with it's own set of axes and everything, and combined with the primitives of circ. Not what you wanted
Final comment. ListPlot
if it's given a one dimensional list, interprets all the numbers as y-coordinates. If you want it to plot (x,y) pairs, you need to give it a list of (x, y) pairs. In your case, a list of only one pair, {{10, 10}}
Hope that now you not only understand why your code didn't work but also why what works works