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The Equiangular Hexagon (Posted on 2014-02-17) Difficulty: 3 of 5
An equiangular hexagon has side lengths 6, 7, 8, 9, 10, 11 (not necessarily in this order). If the area of the hexagon is k√3 , find the sum of all possible values of k.

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution computer solution Comment 1 of 1

DECLARE SUB permute (a$)
CLS
DEFDBL A-Z
pi = ATN(1) * 4
a$ = "6789ab": h$ = a$
DO
  x = 0: y = 0: a = 0
  psn = 0
  FOR angle = 0 TO 300 STEP 60
   psn = psn + 1
   l = INSTR("123456789ab", MID$(a$, psn, 1))
   nx = x + l * COS(angle * pi / 180)
   ny = y + l * SIN(angle * pi / 180)
   a = a + (x - nx) * (y + ny) / 2
   x = nx
   y = ny
  NEXT angle
  IF ABS(x) <= .000001 AND ABS(y) <= .000001 THEN
    PRINT a$, a, a * 4 / SQR(3)
  END IF
  permute a$
LOOP UNTIL a$ = h$

finds the orders for the sides so as to form a closed hexagon, and the area of each:

order of             
lengths
of sides             area                      k*4
(a=10
 b=11)
69a78b         184.0303983041932           424.9999999999999
6a897b         184.8964237079776           427
6b798a         184.8964237079777           427.0000000000001
6b87a9         184.0303983041932           425.0000000000001
78b69a         184.0303983041932           425.0000000000001
798a6b         184.8964237079777           427.0000000000001
7a96b8         184.0303983041932           425.0000000000001
7b6a89         184.8964237079776           427
87a96b         184.0303983041933           425.0000000000001
897b6a         184.8964237079777           427.0000000000001
8a6b79         184.8964237079777           427.0000000000001
8b69a7         184.0303983041933           425.0000000000001
96b87a         184.0303983041932           425.0000000000001
97b6a8         184.8964237079776           427
98a6b7         184.8964237079777           427.0000000000001
9a78b6         184.0303983041933           425.0000000000001
a6b798         184.8964237079777           427.0000000000001
a78b69         184.0303983041933           425.0000000000001
a897b6         184.8964237079777           427.0000000000001
a96b87         184.0303983041933           425.0000000000001
b69a78         184.0303983041932           425.0000000000001
b6a897         184.8964237079777           427.0000000000001
b798a6         184.8964237079777           427.0000000000001
b87a96         184.0303983041932           425.0000000000001

so 427/4 and 425/4 are the two possible values of k.

The sum of these two values is 213.


  Posted by Charlie on 2014-02-17 17:32:03
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