To be honest, I'm now a little stunned myself at how complicated the answer seemed at first, and how simple the solution is in the end.

Here we go:

1) First create your points with `Distribute Points on Faces`.

    [![How could I get the distance of a point to its nearest point - Step 1][1]][1]

2) Then use the node `Convex Hull` to connect the points with a hull.

    [![How could I get the distance of a point to its nearest point - Step 2][2]][2]

3) Using the nodes `Extrude Mesh`, `Split Edges` and `Separate Geometry` you get the isolated edges of this mesh.

    [![How could I get the distance of a point to its nearest point - Step 3][3]][3]

4) Then reduce the scale of each edge by half.

    [![How could I get the distance of a point to its nearest point - Step 4][4]][4]

5) Now that the edges are reduced in proportion to their length, you can reliably find the nearest point with the node `Geometry Proximity` and remember it with `Capture Attribute`.

6) Next, create a couple of curves at each of the previously created points and use `Realize Instances` so that you can influence the endpoints of the curves.

    [![How could I get the distance of a point to its nearest point - Step 6][5]][5]

7) If you then calculate the direction vector between your originally created points and the position results of `Geometry Proximity`, you will know in which direction your new curves should point.

    [![How could I get the distance of a point to its nearest point - Step 7][6]][6]

8) In the last step you only have to correct the length. Since you have shortened the edges by *50%* before, you simply scale the direction vector three times, which is exactly the poiint you were looking for.

    [![How could I get the distance of a point to its nearest point - Step 8][7]][7]

The final result is this:

[![How could I get the distance of a point to its nearest point - Result][8]][8]

Here is an overview of the complete node group:

[![How could I get the distance of a point to its nearest point - Node Group][9]][9]

Here is the blend file:

[<img src="https://blend-exchange.com/embedImage.png?bid=zGjOXRyB" />](https://blend-exchange.com/b/zGjOXRyB/)


  [1]: https://i.sstatic.net/gloUr.png
  [2]: https://i.sstatic.net/KthGW.png
  [3]: https://i.sstatic.net/J4uOG.jpg
  [4]: https://i.sstatic.net/eOtoi.png
  [5]: https://i.sstatic.net/IPZdm.png
  [6]: https://i.sstatic.net/oHKVv.png
  [7]: https://i.sstatic.net/Ybn2S.png
  [8]: https://i.sstatic.net/5w8pl.png
  [9]: https://i.sstatic.net/QiZvn.png