Here are some very close approximations to a full solution to this puzzle. I know that "close" only counts in horseshoes, but a lot of mathematics is approximations, like the value of pi. (Maybe I'll put that in the problem queue: "Find the exact value of pi".) 

 

For distances not distinct:

Radius = 11

Points:

1: (11,0)

2: (5.5, 9.5)

3: (-5.5, 9.5)

4: (-11, 0)

5: (-5.5, -9.5)

6: (5.5, -9.5)

Distances between points:

1-2  11

1-3  19

1-4  22

1-5  19

1-6  11

2-3  11

2-4  19

2-5  22

2-6  19

3-4  11

3-5  19

3-6  22

4-5  11

4-6  19

5-6  11

 

For distinct distances:

 

Revised Visual Basic program:

 

Imports System
Imports System.IO
Imports System.Text
Imports System.Runtime.InteropServices
Imports System.Math
Module Module1
    Sub Main()
        Randomize()
        Dim strstarttime As String
        Dim strendtime As String
        Dim strend As String
        Dim dblradius As Double
        Dim intwork As Integer
        Dim dblwork As Double
        Dim dbltable(15) As Double
        Dim dblcounter As Double = 0
        Dim dbldistance12 As Double
        Dim dbldistance13 As Double
        Dim dbldistance14 As Double
        Dim dbldistance15 As Double
        Dim dbldistance16 As Double
        Dim dbldistance21 As Double
        Dim dbldistance23 As Double
        Dim dbldistance24 As Double
        Dim dbldistance25 As Double
        Dim dbldistance26 As Double
        Dim dbldistance34 As Double
        Dim dbldistance35 As Double
        Dim dbldistance36 As Double
        Dim dbldistance45 As Double
        Dim dbldistance46 As Double
        Dim dbldistance56 As Double
        Dim intpoints As Integer = 0
        Dim dblx1 As Double
        Dim dbly1 As Double
        Dim dblx2 As Double
        Dim dbly2 As Double
        Dim dblx3 As Double
        Dim dbly3 As Double
        Dim dblx4 As Double
        Dim dbly4 As Double
        Dim dblx5 As Double
        Dim dbly5 As Double
        Dim dblx6 As Double
        Dim dbly6 As Double
        strstarttime = TimeOfDay
        Console.WriteLine("Start of execution: " & strstarttime)
        For index1 As Integer = 0 To 14
            dbltable(index1) = 0
        Next
        dblradius = 11
        While dblradius < 99999999 And intpoints < 6
            dblradius += 0.1
            dblwork = Int(dblradius * (10 ^ 2) + 0.5) / _
            (10 ^ 2)
            dblradius = dblwork
            dblx1 = dblradius
            dbly1 = 0
            Console.WriteLine( _
            TimeOfDay & ": Radius=" & Str(dblradius))
            findpoints(dblradius, intpoints, dblcounter, _
            dblx1, dbly1, dblx2, dbly2, _
            dblx3, dbly3, dblx4, dbly4, _
            dblx5, dbly5, dblx6, dbly6, _
            dbldistance12, dbldistance13, _
            dbldistance14, dbldistance15, _
            dbldistance16, dbldistance23, _
            dbldistance24, dbldistance25, _
            dbldistance26, dbldistance34, _
            dbldistance35, dbldistance36, _
            dbldistance45, dbldistance46, _
            dbldistance56, dbltable)
        End While
        strendtime = TimeOfDay
        Console.WriteLine(" ")
        Console.WriteLine("Start of execution: " & strstarttime)
        strendtime = TimeOfDay
        Console.WriteLine("End of execution: " & strendtime)
        strend = "?"
        Console.WriteLine(" ")
        While strend <> "X"
            Console.WriteLine("Please enter X to exit program...")
            strend = UCase(Console.ReadLine())
        End While
    End Sub

    Sub findpoints(ByRef dblradius, ByRef intpoints, _
    ByRef dblcounter, _
    ByRef dblx1, ByRef dbly1, ByRef dblx2, ByRef dbly2, _
    ByRef dblx3, ByRef dbly3, ByRef dblx4, ByRef dbly4, _
    ByRef dblx5, ByRef dbly5, ByRef dblx6, ByRef dbly6, _
    ByRef dbldistance12, ByRef dbldistance13, _
    ByRef dbldistance14, ByRef dbldistance15, _
    ByRef dbldistance16, ByRef dbldistance23, _
    ByRef dbldistance24, ByRef dbldistance25, _
    ByRef dbldistance26, ByRef dbldistance34, _
    ByRef dbldistance35, ByRef dbldistance36, _
    ByRef dbldistance45, ByRef dbldistance46, _
    ByRef dbldistance56, ByRef dbltable)
        Dim dblwork As Double
        dblx2 = dblx1
        While dblx2 > -dblradius
            dblx2 -= 0.1
            findy(dblradius, _
            dblx2, dbly2)
            For index1 As Integer = 0 To 14
                dbltable(index1) = 0
            Next
            findpoint2(dblradius, intpoints, dblcounter, _
            dblx1, dbly1, dblx2, dbly2, _
            dblx3, dbly3, dblx4, dbly4, _
            dblx5, dbly5, dblx6, dbly6, _
            dbldistance12, dbldistance13, _
            dbldistance14, dbldistance15, _
            dbldistance16, dbldistance23, _
            dbldistance24, dbldistance25, _
            dbldistance26, dbldistance34, _
            dbldistance35, dbldistance36, _
            dbldistance45, dbldistance46, _
            dbldistance56, dbltable)
        End While
        If intpoints < 6 Then
            While dblx2 < dblradius
                dblx2 += 0.1
                findy(dblradius, _
                dblx2, dbly2)
                dbly2 *= -1
                findpoint2(dblradius, intpoints, dblcounter, _
                dblx1, dbly1, dblx2, dbly2, _
                dblx3, dbly3, dblx4, dbly4, _
                dblx5, dbly5, dblx6, dbly6, _
                dbldistance12, dbldistance13, _
                dbldistance14, dbldistance15, _
                dbldistance16, dbldistance23, _
                dbldistance24, dbldistance25, _
                dbldistance26, dbldistance34, _
                dbldistance35, dbldistance36, _
                dbldistance45, dbldistance46, _
                dbldistance56, dbltable)
            End While
        End If
    End Sub

    Sub findpoint2(ByRef dblradius, ByRef intpoints, _
    ByRef dblcounter, _
    ByRef dblx1, ByRef dbly1, ByRef dblx2, ByRef dbly2, _
    ByRef dblx3, ByRef dbly3, ByRef dblx4, ByRef dbly4, _
    ByRef dblx5, ByRef dbly5, ByRef dblx6, ByRef dbly6, _
    ByRef dbldistance12, ByRef dbldistance13, _
    ByRef dbldistance14, ByRef dbldistance15, _
    ByRef dbldistance16, ByRef dbldistance23, _
    ByRef dbldistance24, ByRef dbldistance25, _
    ByRef dbldistance26, ByRef dbldistance34, _
    ByRef dbldistance35, ByRef dbldistance36, _
    ByRef dbldistance45, ByRef dbldistance46, _
    ByRef dbldistance56, ByRef dbltable)
        Dim dblwork As Double
        Dim dblwork2 As Double
        Dim dblincr As Double
        If dblx2 = dblx1 And dbly2 = dbly1 Then
            Exit Sub
        End If
        pointdistance(dblx1, dbly1, _
                dblx2, dbly2, dbldistance12)
        If dbldistance12 <> Fix(dbldistance12) Then
            Exit Sub
        End If
        dbltable(0) = dbldistance12
        dblx3 = dblx2
        dbly3 = dbly2
        If dbly3 > 0 Then
            While dblx3 > -dblradius
                dblx3 -= 0.1
                findy(dblradius, _
                dblx3, dbly3)
                findpoint3(dblradius, intpoints, dblcounter, _
                dblx1, dbly1, dblx2, dbly2, _
                dblx3, dbly3, dblx4, dbly4, _
                dblx5, dbly5, dblx6, dbly6, _
                dbldistance12, dbldistance13, _
                dbldistance14, dbldistance15, _
                dbldistance16, dbldistance23, _
                dbldistance24, dbldistance25, _
                dbldistance26, dbldistance34, _
                dbldistance35, dbldistance36, _
                dbldistance45, dbldistance46, _
                dbldistance56, dbltable)
            End While
        End If
        If intpoints < 6 Then
            While dblx3 < dblradius
                dblx3 += 0.1
                findy(dblradius, _
                dblx3, dbly3)
                dbly3 *= -1
                findpoint3(dblradius, intpoints, dblcounter, _
                dblx1, dbly1, dblx2, dbly2, _
                dblx3, dbly3, dblx4, dbly4, _
                dblx5, dbly5, dblx6, dbly6, _
                dbldistance12, dbldistance13, _
                dbldistance14, dbldistance15, _
                dbldistance16, dbldistance23, _
                dbldistance24, dbldistance25, _
                dbldistance26, dbldistance34, _
                dbldistance35, dbldistance36, _
                dbldistance45, dbldistance46, _
                dbldistance56, dbltable)
            End While
        End If
    End Sub

    Sub findpoint3(ByRef dblradius, ByRef intpoints, _
    ByRef dblcounter, _
    ByRef dblx1, ByRef dbly1, ByRef dblx2, ByRef dbly2, _
    ByRef dblx3, ByRef dbly3, ByRef dblx4, ByRef dbly4, _
    ByRef dblx5, ByRef dbly5, ByRef dblx6, ByRef dbly6, _
    ByRef dbldistance12, ByRef dbldistance13, _
    ByRef dbldistance14, ByRef dbldistance15, _
    ByRef dbldistance16, ByRef dbldistance23, _
    ByRef dbldistance24, ByRef dbldistance25, _
    ByRef dbldistance26, ByRef dbldistance34, _
    ByRef dbldistance35, ByRef dbldistance36, _
    ByRef dbldistance45, ByRef dbldistance46, _
    ByRef dbldistance56, ByRef dbltable)
        Dim dblwork As Double
        Dim dblwork2 As Double
        Dim dblincr As Double
        If dblx3 = dblx1 And dbly3 = dbly1 Then
            Exit Sub
        End If
        pointdistance(dblx1, dbly1, _
                dblx3, dbly3, dbldistance13)
        pointdistance(dblx2, dbly2, _
               dblx3, dbly3, dbldistance23)
        If dbldistance13 <> Fix(dbldistance13) Or _
        dbldistance23 <> Fix(dbldistance23) Then
            Exit Sub
        End If
        dbltable(1) = dbldistance13
        dbltable(2) = dbldistance23
        For index1 As Integer = 0 To 1
            For index2 As Integer = index1 + 1 To 2
                If dbltable(index1) = dbltable(index2) And _
                dbltable(index1) <> 0 Then
                    Exit Sub
                End If
            Next
        Next
        dblx4 = dblx3
        dbly4 = dbly3
        If dbly4 > 0 Then
            While dblx4 > -dblradius
                dblx4 -= 0.1
                findy(dblradius, _
                dblx4, dbly4)
                findpoint4(dblradius, intpoints, dblcounter, _
                dblx1, dbly1, dblx2, dbly2, _
                dblx3, dbly3, dblx4, dbly4, _
                dblx5, dbly5, dblx6, dbly6, _
                dbldistance12, dbldistance13, _
                dbldistance14, dbldistance15, _
                dbldistance16, dbldistance23, _
                dbldistance24, dbldistance25, _
                dbldistance26, dbldistance34, _
                dbldistance35, dbldistance36, _
                dbldistance45, dbldistance46, _
                dbldistance56, dbltable)
            End While
        End If
        If intpoints < 6 Then
            While dblx4 < dblradius
                dblx4 += 0.1
                findy(dblradius, _
                dblx4, dbly4)
                dbly4 *= -1
                findpoint4(dblradius, intpoints, dblcounter, _
                dblx1, dbly1, dblx2, dbly2, _
                dblx3, dbly3, dblx4, dbly4, _
                dblx5, dbly5, dblx6, dbly6, _
                dbldistance12, dbldistance13, _
                dbldistance14, dbldistance15, _
                dbldistance16, dbldistance23, _
                dbldistance24, dbldistance25, _
                dbldistance26, dbldistance34, _
                dbldistance35, dbldistance36, _
                dbldistance45, dbldistance46, _
                dbldistance56, dbltable)
            End While
        End If
    End Sub

    Sub findpoint4(ByRef dblradius, ByRef intpoints, _
        ByRef dblcounter, _
        ByRef dblx1, ByRef dbly1, ByRef dblx2, ByRef dbly2, _
        ByRef dblx3, ByRef dbly3, ByRef dblx4, ByRef dbly4, _
        ByRef dblx5, ByRef dbly5, ByRef dblx6, ByRef dbly6, _
        ByRef dbldistance12, ByRef dbldistance13, _
        ByRef dbldistance14, ByRef dbldistance15, _
        ByRef dbldistance16, ByRef dbldistance23, _
        ByRef dbldistance24, ByRef dbldistance25, _
        ByRef dbldistance26, ByRef dbldistance34, _
        ByRef dbldistance35, ByRef dbldistance36, _
        ByRef dbldistance45, ByRef dbldistance46, _
        ByRef dbldistance56, ByRef dbltable)
        Dim dblwork As Double
        Dim dblwork2 As Double
        Dim dblincr As Double
        If dblx4 = dblx1 And dbly4 = dbly1 Then
            Exit Sub
        End If
        pointdistance(dblx1, dbly1, _
                dblx4, dbly4, dbldistance14)
        pointdistance(dblx2, dbly2, _
               dblx4, dbly4, dbldistance24)
        pointdistance(dblx3, dbly3, _
              dblx4, dbly4, dbldistance34)
        If dbldistance14 <> Fix(dbldistance14) Or _
        dbldistance24 <> Fix(dbldistance24) Or _
        dbldistance34 <> Fix(dbldistance34) Then
            Exit Sub
        End If
        dbltable(3) = dbldistance14
        dbltable(4) = dbldistance24
        dbltable(5) = dbldistance34
        For index1 As Integer = 0 To 4
            For index2 As Integer = index1 + 1 To 5
                If dbltable(index1) = dbltable(index2) And _
                dbltable(index1) <> 0 Then
                    Exit Sub
                End If
            Next
        Next
        dblx5 = dblx4
        dbly5 = dbly4
        If dbly5 > 0 Then
            While dblx5 > -dblradius
                dblx5 -= 0.1
                findy(dblradius, _
                dblx5, dbly5)
                findpoint5(dblradius, intpoints, dblcounter, _
                dblx1, dbly1, dblx2, dbly2, _
                dblx3, dbly3, dblx4, dbly4, _
                dblx5, dbly5, dblx6, dbly6, _
                dbldistance12, dbldistance13, _
                dbldistance14, dbldistance15, _
                dbldistance16, dbldistance23, _
                dbldistance24, dbldistance25, _
                dbldistance26, dbldistance34, _
                dbldistance35, dbldistance36, _
                dbldistance45, dbldistance46, _
                dbldistance56, dbltable)
            End While
        End If
        If intpoints < 6 Then
            While dblx5 < dblradius
                dblx5 += 0.1
                findy(dblradius, _
                dblx5, dbly5)
                dbly5 *= -1
                findpoint5(dblradius, intpoints, dblcounter, _
                dblx1, dbly1, dblx2, dbly2, _
                dblx3, dbly3, dblx4, dbly4, _
                dblx5, dbly5, dblx6, dbly6, _
                dbldistance12, dbldistance13, _
                dbldistance14, dbldistance15, _
                dbldistance16, dbldistance23, _
                dbldistance24, dbldistance25, _
                dbldistance26, dbldistance34, _
                dbldistance35, dbldistance36, _
                dbldistance45, dbldistance46, _
                dbldistance56, dbltable)
            End While
        End If
    End Sub

    Sub findpoint5(ByRef dblradius, ByRef intpoints, _
        ByRef dblcounter, _
        ByRef dblx1, ByRef dbly1, ByRef dblx2, ByRef dbly2, _
        ByRef dblx3, ByRef dbly3, ByRef dblx4, ByRef dbly4, _
        ByRef dblx5, ByRef dbly5, ByRef dblx6, ByRef dbly6, _
        ByRef dbldistance12, ByRef dbldistance13, _
        ByRef dbldistance14, ByRef dbldistance15, _
        ByRef dbldistance16, ByRef dbldistance23, _
        ByRef dbldistance24, ByRef dbldistance25, _
        ByRef dbldistance26, ByRef dbldistance34, _
        ByRef dbldistance35, ByRef dbldistance36, _
        ByRef dbldistance45, ByRef dbldistance46, _
        ByRef dbldistance56, ByRef dbltable)
        Dim dblwork As Double
        Dim dblwork2 As Double
        Dim dblincr As Double
        If dblx5 = dblx1 And dbly5 = dbly1 Then
            Exit Sub
        End If
        pointdistance(dblx1, dbly1, _
                dblx5, dbly5, dbldistance15)
        pointdistance(dblx2, dbly2, _
               dblx5, dbly5, dbldistance25)
        pointdistance(dblx3, dbly3, _
              dblx5, dbly5, dbldistance35)
        pointdistance(dblx4, dbly4, _
              dblx5, dbly5, dbldistance45)
        If dbldistance15 <> Fix(dbldistance15) Or _
        dbldistance25 <> Fix(dbldistance25) Or _
        dbldistance35 <> Fix(dbldistance35) Or _
        dbldistance45 <> Fix(dbldistance45) Then
            Exit Sub
        End If
        dbltable(6) = dbldistance15
        dbltable(7) = dbldistance25
        dbltable(8) = dbldistance35
        dbltable(9) = dbldistance45
        For index1 As Integer = 0 To 8
            For index2 As Integer = index1 + 1 To 9
                If dbltable(index1) = dbltable(index2) And _
                dbltable(index1) <> 0 Then
                    Exit Sub
                End If
            Next
        Next
        dblx6 = dblx5
        dbly6 = dbly5
        If dbly6 > 0 Then
            While dblx6 > -dblradius
                dblx6 -= 0.1
                findy(dblradius, _
                dblx6, dbly6)
                findpoint6(dblradius, intpoints, dblcounter, _
                dblx1, dbly1, dblx2, dbly2, _
                dblx3, dbly3, dblx4, dbly4, _
                dblx5, dbly5, dblx6, dbly6, _
                dbldistance12, dbldistance13, _
                dbldistance14, dbldistance15, _
                dbldistance16, dbldistance23, _
                dbldistance24, dbldistance25, _
                dbldistance26, dbldistance34, _
                dbldistance35, dbldistance36, _
                dbldistance45, dbldistance46, _
                dbldistance56, dbltable)
            End While
        End If
        If intpoints < 6 Then
            While dblx6 < dblradius
                dblx6 += 0.1
                findy(dblradius, _
                dblx6, dbly6)
                dbly6 *= -1
                findpoint6(dblradius, intpoints, dblcounter, _
                dblx1, dbly1, dblx2, dbly2, _
                dblx3, dbly3, dblx4, dbly4, _
                dblx5, dbly5, dblx6, dbly6, _
                dbldistance12, dbldistance13, _
                dbldistance14, dbldistance15, _
                dbldistance16, dbldistance23, _
                dbldistance24, dbldistance25, _
                dbldistance26, dbldistance34, _
                dbldistance35, dbldistance36, _
                dbldistance45, dbldistance46, _
                dbldistance56, dbltable)
            End While
        End If
    End Sub

 

    Sub findpoint6(ByRef dblradius, ByRef intpoints, _
    ByRef dblcounter, _
    ByRef dblx1, ByRef dbly1, ByRef dblx2, ByRef dbly2, _
    ByRef dblx3, ByRef dbly3, ByRef dblx4, ByRef dbly4, _
    ByRef dblx5, ByRef dbly5, ByRef dblx6, ByRef dbly6, _
    ByRef dbldistance12, ByRef dbldistance13, _
    ByRef dbldistance14, ByRef dbldistance15, _
    ByRef dbldistance16, ByRef dbldistance23, _
    ByRef dbldistance24, ByRef dbldistance25, _
    ByRef dbldistance26, ByRef dbldistance34, _
    ByRef dbldistance35, ByRef dbldistance36, _
    ByRef dbldistance45, ByRef dbldistance46, _
    ByRef dbldistance56, ByRef dbltable)
        If dblx6 = dblx1 And dbly6 = dbly1 Then
            Exit Sub
        End If
        pointdistance(dblx1, dbly1, _
                dblx6, dbly6, dbldistance16)
        pointdistance(dblx2, dbly2, _
               dblx6, dbly6, dbldistance26)
        pointdistance(dblx3, dbly3, _
              dblx6, dbly6, dbldistance36)
        pointdistance(dblx4, dbly4, _
              dblx6, dbly6, dbldistance46)
        pointdistance(dblx5, dbly5, _
              dblx6, dbly6, dbldistance56)
        If dbldistance16 <> Fix(dbldistance16) Or _
        dbldistance26 <> Fix(dbldistance26) Or _
        dbldistance36 <> Fix(dbldistance36) Or _
        dbldistance46 <> Fix(dbldistance46) Or _
        dbldistance56 <> Fix(dbldistance56) Then
            Exit Sub
        End If
        dbltable(10) = dbldistance16
        dbltable(11) = dbldistance26
        dbltable(12) = dbldistance36
        dbltable(13) = dbldistance46
        dbltable(14) = dbldistance56
        For index1 As Integer = 0 To 13
            For index2 As Integer = index1 + 1 To 14
                If dbltable(index1) = dbltable(index2) And _
                dbltable(index1) <> 0 Then
                    Exit Sub
                End If
            Next
        Next
        intpoints = 6
        Console.WriteLine("Eureka !!!!")
        Console.WriteLine("Radius=" & Str(dblradius))
        Console.WriteLine("The six points are:")
        Console.WriteLine( _
        "(" & Str(dblx1) & "," & Str(dbly1) & ")")
        Console.WriteLine( _
        "(" & Str(dblx2) & "," & Str(dbly2) & ")")
        Console.WriteLine( _
        "(" & Str(dblx3) & "," & Str(dbly3) & ")")
        Console.WriteLine( _
        "(" & Str(dblx4) & "," & Str(dbly4) & ")")
        Console.WriteLine( _
        "(" & Str(dblx5) & "," & Str(dbly5) & ")")
        Console.WriteLine( _
        "(" & Str(dblx6) & "," & Str(dbly6) & ")")
        Console.WriteLine("Distances:")
        Console.WriteLine( _
        "Points 1-2: " & Str(dbldistance12))
        Console.WriteLine( _
        "Points 1-3: " & Str(dbldistance13))
        Console.WriteLine( _
        "Points 1-4: " & Str(dbldistance14))
        Console.WriteLine( _
        "Points 1-5: " & Str(dbldistance15))
        Console.WriteLine( _
        "Points 1-6: " & Str(dbldistance16))
        Console.WriteLine( _
        "Points 2-3: " & Str(dbldistance23))
        Console.WriteLine( _
        "Points 2-4: " & Str(dbldistance24))
        Console.WriteLine( _
        "Points 2-5: " & Str(dbldistance25))
        Console.WriteLine( _
        "Points 2-6: " & Str(dbldistance26))
        Console.WriteLine( _
        "Points 3-4: " & Str(dbldistance34))
        Console.WriteLine( _
        "Points 3-5: " & Str(dbldistance35))
        Console.WriteLine( _
        "Points 3-6: " & Str(dbldistance36))
        Console.WriteLine( _
        "Points 4-5: " & Str(dbldistance45))
        Console.WriteLine( _
        "Points 4-6: " & Str(dbldistance46))
        Console.WriteLine( _
        "Points 5-6: " & Str(dbldistance56))
        Console.WriteLine("Please press enter to continue.")
        Console.ReadLine()
    End Sub

    Sub findy(ByRef dblradius, _
    ByRef dblx, ByRef dbly)
        Dim dblwork As Double
        dblwork = Int(dblx * (10 ^ 1) + 0.5) / _
                    (10 ^ 1)
        dblx = dblwork
        dbly = Sqrt((dblradius ^ 2) - (dblx ^ 2))
        If Double.IsNaN(dbly) Then
            dbly = 0
        End If
        dblwork = Int(dbly * (10 ^ 1) + 0.5) / _
        (10 ^ 1)
        dbly = dblwork
    End Sub

    Sub pointdistance(ByRef dblxa, ByRef dblya, _
    ByRef dblxb, ByRef dblyb, ByRef dbldistance)
        Dim dblwork As Double
        dblwork = _
        ((dblxa - dblxb) ^ 2) + ((dblya - dblyb) ^ 2)
        dbldistance = Sqrt(dblwork)
        If Double.IsNaN(dbldistance) Then
            dbldistance = 0
        End If
        dblwork = Int(dbldistance * (10 ^ 1) + 0.5) / _
        (10 ^ 1)
        dbldistance = dblwork
    End Sub
End Module