Magic Triangles in a Hexagon

    A - B - C
   / \ / \ / \
  D - E - F - G
 / \ / \ / \ / \
H - I - J - K - L
 \ / \ / \ / \ /
  M - N - O - P
   \ / \ / \ /
    Q - R - S
Place the numbers 1-19 in the hexagon, such that each of the twelve triangles of six numbers have the same sum. These include triangles such as ABCEFJ and BEFIJK. What are the lowest and highest possible common sums?

Extension: Find a solution which also has these sets equal to the common sum: the six corners (ACHLQS), the six sides (BDGMPR), and the six central values (EFIKNO).

For ease in comparing solutions, place the lowest-valued corner at A, and the lowest adjacent side at B. I know a computer can help solve this, but I'd be interested in any analytical approaches that can be found.

Source: Original.

Solutions were received from Philippe Fondanaiche and Sudipta Das.  An example solution for each possible sum is below, followed by Philippe's analytical approach and Sudipta's exhaustive list of solutions.  Only the sum of 60 completely solves the Extension puzzle.
Sum = 50
    1 12 17
 19 13  5  9
 8  7  2 11  6
 14  4 10 18
   15 16  3
 
Sum = 52
    2 12 13
 17 15  4 10
11  1  6 14  5
 18  9  3 16
    7 19  8
 
Sum = 53
    1  8 19
  9 18  5 11
16  7  2 13  3
 10 12  4 17
    6 15 14
 
Sum = 54
    1 10 18
 19 16  5  3
12  2  4 17  7
 15  8  9 11
   13 14  6
 
Sum = 55
    1  5 19
 16 14 10  3
11  7  6 13  4
 17 2 9 15
   12 18  8
Sum = 56
    1  4 18
  6 16 15  3
19 12  2  7 11
  8  5 13 14
   10 17 9
 
Sum = 57
    1  7 15
 12 16 10  4
18  2  8 14  6
 17  9  5 11
    3 19 13
 
Sum = 58
    1  7 14
  8 13 17  2
19 11  6  4 15
  9  3 16 12
   10 18 5
 
Sum = 59
    1  4 19
 14 15  8  5
10  7 12 13  2
 16  3  6 17
   11 18 9
 
Sum = 60
    1  4 19
  6 12 14  8
18 13 10  7  2
 11  5  9 15
    3 16 17
Sum = 61
    1  2 19
  4 17 14  3
18 13  8  7 10
  6  5 12 15
   11 16  9
 
Sum = 62
    1  8 19
 18 16  4  2
10  3 14 17  6
 13  7  9 11
   15 12  5
 
Sum = 63
    2  1 17
  4 16 15  3
19 10 12  9  7
 11  6  8 13
    5 18 14
 
Sum = 64
    1  6 19
 17 13  7  3
10  5 18 15 2
 16  4  8 12
   11 14  9
 
Sum = 65
    1  2 19
  3 18 11  5
16 13 14  7  9
  4  6 10 17
   12 15  8
Sum = 66
    2  5 13
  6 12 18  1
19 11 16  4 14
  9  3 15 10
    8 17  7
 
Sum = 67
    1  5 19
 10  8 16  3
17 13 18  7  4
 11  2 15  9
    6 12 14
 
Sum = 68
    7  1 18
  2 11 17  4
15 19 14  6  9
  3  5 16 10
   12  8 13
 
Sum = 70
    3  4 19
  7 15 11  1
14 13 18  9 12
  2  6 16 10
   17  8  5
Philippe's Analysis:
Let SU the common sum to the twelve triangles, S1 the sum of the six corners, S2 the sum of the six sides and S3 the sum of the six central values.
We have the two following identities:
 - the first one is obtained by adding all the terms of the hexagon:
   S1+S2+S3+J = 1+2+3+....+19 = 190
   Considering the 12 equations such as A+B+C+E+F+J=SU, A+D+E+H+I+J,etc..,it is easy to check that S1=S3.
   So the equation becomes 2*S1+S2+J=190
 - the second one is obtained by adding the twelve equations:
   6*SU = 2*S3 +5*J +190
As a consequence J is always even.
The lowest theoretical value of SU is got for J=2 and S3=1+3+4+5+6+7=21,that is to say SU min=48,
whereas the highest theoretical value of SU is got for J=18 and S3=13+14+15+16+17+19=94,so SU max=78.

We have a system of 12 equations with 2 unknown variables  (A to S + SU) which are all different.

In a first step let consider the case S1=S3=SU
Then 4*SU=5*J+190 and the possible values of J and SU are:
J=2 SU=50
J=6 SU=55
J=10 SU=60. In this case we can check that S1=S2=S3=SU=60
J=14 SU=65
J=18 SU=70
For each couple (SU,J), it is very easy with the help of a computer and relatively easy manually to find solutions. Indeed the 12 initial equations can be expressed
more simply under the following identities:
A+C = I+K = Q+S
A+H = F+N = L+S
C+L = E+O = H+Q
B+I = G+O
B+K = D+N
E+G = N+P
F+P = I+R
K+R = E+M
D+F = M+O

Sudipta Das' list of solutions:

Here is a list of solutions : 

Sum    Number of solutions
50                 6
52                 2
53                 4
54                 2
55                22
56                12
57                 8
58                 4
59                16
60                 8
Another 84 solutions can be obtained from those listed above by replacing each number N by ( 20 - N ). So, from a magic triangled-hexagon with a sum S, you can get a magic triangled-hexagon with a sum of (20 * 6 - S) or (120 - S). The first 84 solutions are: 
Read the solutions as:
Sum : A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S
* : Sum of corners = Common Sum
+ : Sum of sides = Common Sum
# : Sum of center six = Common Sum

50: 1,12,17,18,14,4,10,8,7,2,11,6,13,5,9,19,15,16,3 *#
50: 1,12,17,19,13,5,9,8,7,2,11,6,14,4,10,18,15,16,3 *#
50: 1,16,17,18,10,4,14,8,11,2,7,6,9,5,13,19,15,12,3 *#
50: 1,16,17,19,9,5,13,8,11,2,7,6,10,4,14,18,15,12,3 *#
50: 5,12,13,19,15,3,9,8,1,2,17,6,18,10,4,14,11,16,7 *#
50: 5,14,8,16,4,17,9,13,10,2,3,11,18,1,15,12,6,19,7 *#

52: 2,12,13,17,15,4,10,11,1,6,14,5,18,9,3,16,7,19,8
52: 2,16,11,19,3,14,10,13,9,6,4,7,18,1,15,12,5,17,8

53: 1,8,19,9,18,5,11,16,7,2,13,3,10,12,4,17,6,15,14
53: 1,15,19,17,4,12,10,16,13,2,7,3,11,5,18,9,6,8,14
53: 4,13,12,19,14,2,11,7,1,8,15,5,18,9,3,16,10,17,6
53: 4,16,7,17,3,15,11,12,9,8,2,10,18,1,14,13,5,19,6

54: 1,10,18,19,16,5,3,12,2,4,17,7,15,8,9,11,13,14,6
54: 1,11,12,14,9,17,3,18,8,4,5,13,15,2,16,10,7,19,6

55: 1,5,19,16,14,10,3,11,7,6,13,4,17,2,9,15,12,18,8 *#
55: 1,6,19,8,11,14,9,18,13,4,7,2,12,5,10,15,3,16,17
55: 1,7,18,8,10,15,9,19,13,4,6,3,12,5,11,14,2,16,17
55: 1,8,18,17,9,13,4,15,7,6,12,2,19,3,11,10,5,16,14 *#
55: 1,9,19,16,18,2,7,11,3,6,17,4,13,10,5,15,12,14,8 *#
55: 1,10,15,16,11,12,4,18,3,6,13,5,19,7,9,8,2,17,14 *#
55: 1,11,18,19,16,3,5,9,4,6,15,8,12,7,10,14,17,13,2 *#
55: 1,13,18,14,10,7,12,9,15,6,4,8,5,3,16,19,17,11,2 *#
55: 1,14,19,15,5,10,13,11,17,6,3,4,7,2,18,16,12,9,8 *#
55: 1,15,11,18,9,13,3,19,2,6,10,12,17,7,14,5,4,16,8 *#
55: 1,15,18,16,10,7,9,19,5,4,14,3,12,13,11,6,2,8,17
55: 1,14,19,16,11,6,9,18,5,4,15,2,12,13,10,7,3,8,17
55: 2,4,18,15,14,11,3,10,8,6,12,5,17,1,9,16,13,19,7 *#
55: 2,9,18,15,19,1,8,10,3,6,17,5,12,11,4,16,13,14,7 *#
55: 2,14,18,16,4,11,12,10,17,6,3,5,8,1,19,15,13,9,7 *#
55: 2,16,10,19,9,12,3,18,1,6,11,13,17,8,14,4,5,15,7 *#
55: 3,7,14,17,16,9,1,11,2,6,15,10,18,5,8,12,13,19,4 *#
55: 3,7,16,19,13,10,1,9,5,6,14,8,18,2,11,12,15,17,4 *#
55: 3,11,16,19,17,2,5,9,1,6,18,8,14,10,7,12,15,13,4 *#
55: 3,12,9,13,7,18,5,16,10,6,2,15,14,1,17,11,8,19,4 *#
55: 3,12,9,17,11,14,1,16,2,6,10,15,18,5,13,7,8,19,4 *#
55: 3,12,11,19,8,15,1,14,5,6,9,13,18,2,16,7,10,17,4 *#

56: 1,4,18,6,16,15,3,19,12,2,7,11,8,5,13,14,10,17,9
56: 1,6,19,8,12,16,4,18,15,2,5,10,7,3,17,13,11,14,9
56: 1,13,18,14,17,5,4,19,3,2,16,11,7,15,12,6,10,8,9
56: 1,14,19,17,13,7,3,18,5,2,15,10,8,12,16,4,11,6,9
56: 5,1,15,3,16,17,4,18,12,2,8,10,11,6,9,14,7,19,13
56: 5,3,15,8,19,12,1,18,4,2,16,10,14,11,6,9,7,17,13
56: 5,3,18,8,9,19,4,15,17,2,6,7,11,1,16,12,10,14,13
56: 5,4,15,9,19,11,1,18,3,2,17,10,14,12,6,8,7,16,13
56: 5,8,18,16,6,17,1,15,12,2,11,7,14,3,19,4,10,9,13
56: 5,9,18,17,6,16,1,15,11,2,12,7,14,4,19,3,10,8,13
56: 5,12,15,14,16,6,4,18,1,2,19,10,11,17,9,3,7,8,13
56: 5,14,18,19,9,8,4,15,6,2,17,7,11,12,16,1,10,3,13

57: 1,7,15,11,17,9,5,18,2,8,14,6,16,10,4,12,3,19,13
57: 1,7,15,12,16,10,4,18,2,8,14,6,17,9,5,11,3,19,13
57: 1,7,18,9,19,4,10,15,5,8,14,3,11,12,2,17,6,16,13
57: 1,7,18,17,11,12,2,15,5,8,14,3,19,4,10,9,6,16,13
57: 1,9,15,16,10,14,2,18,4,8,12,6,19,5,11,7,3,17,13
57: 1,11,18,19,5,14,4,15,9,8,10,3,17,2,16,7,6,12,13
57: 1,12,18,19,4,14,5,15,10,8,9,3,16,2,17,7,6,11,13
57: 1,16,18,17,2,12,11,15,14,8,5,3,10,4,19,9,6,7,13

58: 1,7,14,8,13,17,2,19,11,6,4,15,9,3,16,12,10,18,5
58: 1,8,19,9,11,13,7,14,17,6,3,10,4,2,18,16,15,12,5
58: 1,12,19,16,18,2,4,14,3,6,17,10,7,13,11,9,15,8,5
58: 1,12,19,18,16,4,2,14,3,6,17,10,9,11,13,7,15,8,5

59: 1,4,19,14,15,8,5,10,7,12,13,2,16,3,6,17,11,18,9
59: 1,8,19,18,15,4,5,10,3,12,17,2,16,7,6,13,11,14,9
59: 1,13,10,14,6,17,5,19,7,12,4,11,16,3,15,8,2,18,9
59: 1,17,10,18,6,13,5,19,3,12,8,11,16,7,15,4,2,14,9
59: 2,6,18,17,12,11,1,14,4,10,16,3,19,5,9,8,7,15,13
59: 2,8,14,12,6,19,4,18,11,10,5,7,16,1,15,9,3,17,13
59: 2,8,14,15,9,16,1,18,5,10,11,7,19,4,12,6,3,17,13
59: 2,9,18,17,15,5,4,14,1,10,19,3,16,11,6,8,7,12,13
59: 5,1,19,2,18,12,6,14,16,4,8,10,3,7,11,17,15,13,9
59: 5,2,14,3,16,18,1,19,12,4,7,15,8,6,13,11,10,17,9
59: 5,4,19,7,17,12,1,15,13,2,11,14,3,8,16,10,18,9,6
59: 5,9,19,10,16,8,3,15,11,2,13,14,1,12,17,7,18,4,6
59: 5,11,19,17,13,7,1,14,6,4,18,10,8,12,16,2,15,3,9
59: 5,13,19,17,11,7,3,14,8,4,16,10,6,12,18,2,15,1,9
59: 7,1,19,3,13,17,4,16,18,2,8,9,5,6,15,11,12,10,14
59: 7,10,19,11,15,6,5,16,8,2,18,9,4,17,13,3,12,1,14

60: 1,4,19,6,12,14,8,18,13,10,7,2,11,5,9,15,3,16,17 *+#
60: 1,4,19,9,15,11,5,18,7,10,13,2,14,8,6,12,3,16,17 *+#
60: 1,6,18,7,9,16,8,19,14,10,5,3,11,4,12,13,2,15,17 *+#
60: 1,7,19,12,15,8,5,18,4,10,16,2,14,11,6,9,3,13,17 *+#
60: 5,1,15,3,12,17,8,14,16,10,4,6,11,2,9,18,7,19,13 *+#
60: 5,1,15,9,18,11,2,14,4,10,16,6,17,8,3,12,7,19,13 *+#
60: 5,3,14,4,9,19,8,15,17,10,2,7,11,1,12,16,6,18,13 *+#
60: 5,4,15,12,18,8,2,14,1,10,19,6,17,11,3,9,7,16,13 *+#
Obviously, there are 16 solutions for the Extension puzzle. 8 of them are shown. The other 8 can be obtained by replacing each number N by ( 20 - N ) .
Mail to Ken