Automatically draw chords from n-1 points on a circle to a fixed point on the circle

Question

I want to put n equally spaced points on a circle and automatically draw chords from all points to a fixed point preferably P(0,0).

I have managed to draw the points and the circle so far with

func[n_] := Graphics[{Orange, Thickness[0.001],
{Circle[{0, 0}, 1], {Darker[Blue], PointSize@0.02, 
  Point[Table[{Cos[a], Sin[a]}, {a, 0, 2 \[Pi] - (2 \[Pi])/n, (
     2 \[Pi])/n}]]}}}];

Manipulate[func[n], {n, 3, 50}]

For n=6 it should somehow look like this

func[n_] := Graphics[{Orange, Thickness[0.001],
{{Circle[{0, 0}, 1], {Darker[Blue], PointSize@0.02, 
   Point[Table[{Cos[a], Sin[a]}, {a, 0, 2 \[Pi] - (2 \[Pi])/n, (
      2 \[Pi])/n}]]}},
 Line[{
   {Cos[(2 \[Pi])/6], Sin[(2 \[Pi])/6]}, {Cos[0], 
    Sin[0]}, {Cos[2 (2 \[Pi])/6], Sin[2 (2 \[Pi])/6]}, {Cos[0], 
    Sin[0]}, {Cos[3 (2 \[Pi])/6], Sin[3 (2 \[Pi])/6]}, {Cos[0], 
    Sin[0]}, {Cos[4 (2 \[Pi])/6], Sin[4 (2 \[Pi])/6]}, {Cos[0], 
    Sin[0]}, {Cos[5 (2 \[Pi])/6], Sin[5 (2 \[Pi])/6]}, {Cos[0], 
    Sin[0]}
   }]}}];

Manipulate[func[n], {n, 3, 50}]

Help is very much appreciated.

Answer 1

You can also use StarGraph to get the desired picture:

Manipulate[StarGraph[n, Prolog -> Circle[], ImagePadding -> 20, PlotRange -> 1], 
  {{n, 7}, 3, 20, 1}]

Specify the vertex coordinates to have all the points on the circle:

Manipulate[StarGraph[n, Prolog -> Circle[], 
  VertexCoordinates -> CirclePoints[n], ImagePadding -> 20, 
  PlotRange -> 1], 
 {{n, 7}, 2, 20, 1}]

We can also use LocatorPane and add/delete nodes using (ALT + Click):

DynamicModule[{pts = CirclePoints[7]}, 
 LocatorPane[Dynamic[pts, (pts = Normalize /@ #) &], 
  Dynamic[StarGraph[Length @ pts, 
    VertexCoordinates -> pts, 
    ImagePadding -> 20, 
    Prolog -> Circle[], 
    GraphStyle -> "ThickEdge", 
    VertexShapeFunction -> (GraphElementData["Star"][#, #2, {1, 1}/15] &),
    PlotRange -> 1]], 
  Appearance -> None, LocatorAutoCreate -> {2, 20}]]

Answer 2

chords[n_Integer?Positive] :=
 Graphics[
  {Orange,
   Circle[],
   GraphicsComplex[CirclePoints[{1, 0}, n],
    {
     Table[Line[{1, k}], {k, 2, n}],
     {Black, PointSize[Large], Point@Range[n]}
     }]
   }
  ]

Manipulate[chords[n], {{n, 6}, 1, 20, 1}]

Answer 3

Since you said the fixed point should be at the origin; however, you can move it elsewhere.

Clear["Global`*"]

Manipulate[
 Graphics[{
   Circle[],
   Blue,
   Line[{pt, #}] & /@ CirclePoints[n]}],
 {{pt, {0, 0}}, Locator},
 {{n, 8}, Range[2, 20]}]

Answer 4

Manipulate[
 PolarPlot[1, {t, 0, 2 Pi}, 
  PolarGridLines -> {Subdivide[0, 2 \[Pi], n], None}, 
  GridLinesStyle -> Directive[Dashed, Orange], 
  PolarAxesOrigin -> {0, 1}, PolarTicks -> {None, None}, 
  PolarAxes -> False, Axes -> False], {n, 1, 20, 1}]