# Trying to study the three points theorem of the Mobius Transformation

I am trying to study the three points theorem of the Mobius transformation. This theorem says that the Mobius transformation $M$ maps three distinct points p, q, r to three distinct points p', q', r'. Based on this formula $$\frac{(z-p)(q-r)}{(z-r)(q-p)}=\frac{(w-p')(q'-r')}{(w-r')(q'-p')}$$, I solved for $w$ and came up with this function.

mobius[z_, z1_, z2_, z3_, w1_, w2_, w3_] :=
(w1 (-w2 (z1 - z2) (z - z3) + w3 (z - z2) (z1 - z3)) -
w2 w3 (z - z1) (z2 - z3))/(w3 (z1 - z2) (z - z3) -
w2 (z - z2) (z1 - z3) + w1 (z - z1) (z2 - z3))


If I substitute z with a eaquation of plane, then I can see how the $M$ maps the plane. What I want to know is that is there a way to mark these points on the graph so that I can see how these points behave?? I am using parametric plot to see the graph. Thanks!

--------Edit---------

plane[x_, y_] := x + y I;
Manipulate[
ParametricPlot[
{{plane[x, y] // Re, plane[x, y] // Im},
{mobius[plane[x, y], z1, z2, z3, w1, w2, w3] //Re,
mobius[plane[x, y], z1, z2, z3, w1, w2, w3] // Im}},
{x, 0, 40}, {y, 0, -40}, PlotRange -> {{-50, 50}, {-50, 50}}],
{z1, {10, 20}}, {z2, {10 I, 20 I}},
{z3, {15, 25}},
{w1, {10, 20}}, {w2, {10 I, 20 I}},
{w3, {15, 25}}]

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@Öskå Editted!!! –  eChung00 May 1 at 17:55
You might want to take a look at Epilog in order to mark the points you are interested in. –  Öskå May 1 at 18:18
@Öskå is it possible to see an example?? I am not good coding in mathematica... –  eChung00 May 1 at 19:36
Which point would you like to mark? –  Öskå May 1 at 19:39
@Öskå The three points in the domain and the three points in the image plane... So what I am trying to do is see how these points in the domain plane mapped into the image plane... –  eChung00 May 1 at 19:58