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Minimum perimeter length (Posted on 2010-12-10) Difficulty: 2 of 5
The length of each of the sides of a triangle ABC is a positive integer with: ∠ BAC = 2* ∠ ABC and, ∠ ACB is obtuse.

Find the minimum length of the perimeter.

No Solution Yet Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re(2): this program finds... (spoiler) | Comment 4 of 5 |
(In reply to re: this program finds... (spoiler) by Jer)

I should have put a check in the program to generate only fundamental solutions. Such a list is:

     sides       peri              angles (deg.)
 16   28   33      77    28.9550244  57.9100487  93.1349269
 25   45   56     126    25.8419328  51.6838655 102.4742017
 36   66   85     187    23.5564643  47.1129286 109.3306071
 49   91  120     260    21.7867893  43.5735786 114.6396321
 64  120  161     345    20.3641348  40.7282696 118.9075956
 81  144  175     400    27.2660445  54.5320889  98.2018666
 81  153  208     442    19.1881365  38.3762729 122.4355906
100  190  261     551    18.1948723  36.3897447 125.4153830
121  220  279     620    24.6199773  49.2399547 106.1400680
121  231  320     672    17.3414428  34.6828856 127.9756716
144  276  385     805    16.5978421  33.1956843 130.2064736
169  299  360     828    27.7957725  55.5915450  96.6126825
169  312  407     888    22.6198649  45.2397299 112.1404052
169  325  456     950    15.9423686  31.8847372 132.1728942
196  350  429     975    26.7655006  53.5310012  99.7034983
225  390  451    1066    29.9264349  59.8528697  90.2206954
196  378  533    1107    15.3588856  30.7177712 133.9233433
225  420  559    1204    21.0394698  42.0789396 116.8815907
225  435  616    1276    14.8351116  29.6702232 135.4946653
256  464  585    1305    25.0078332  50.0156665 104.9765003
289  510  611    1410    28.0724869  56.1449739  95.7825392
256  496  705    1457    14.3615116  28.7230231 136.9154653
289  527  672    1488    24.2496286  48.4992572 107.2511142
289  544  735    1568    19.7499228  39.4998456 120.7502316
289  561  800    1650    13.9305546  27.8611091 138.2083363
361  627  728    1716    29.7243291  59.4486583  90.8270126
361  646  795    1802    26.5253520  53.0507040 100.4239440
324  630  901    1855    13.5362027  27.0724055 139.3913918
361  665  864    1890    22.9195422  45.8390843 111.2413735
361  684  935    1980    18.6717181  37.3434361 123.9848458
361  703 1008    2072    13.1735511  26.3471022 140.4793467
400  740  969    2109    22.3316450  44.6632900 113.0050650
441  777  928    2146    28.2425372  56.4850744  95.2723884
441  798 1003    2242    25.2087653  50.4175306 104.3737041
400  780 1121    2301    12.8385681  25.6771363 141.4842956
484  858 1037    2379    27.5801992  55.1603984  97.2594024
441  840 1159    2440    17.7527902  35.5055803 126.7416295
529  920 1071    2520    29.5918458  59.1836916  91.2244626
441  861 1240    2542    12.5279054  25.0558109 142.4162837
484  902 1197    2583    21.2799665  42.5599329 116.1601006
529  943 1152    2624    26.9623890  53.9247780  99.1128330
529  966 1235    2730    24.0706146  48.1412292 107.7881561
484  946 1365    2795    12.2387558  24.4775115 143.2837327
529  989 1320    2838    20.8069478  41.6138956 117.5791567
576 1032 1273    2881    26.3843297  52.7686595 100.8470108
529 1012 1407    2948    16.9574263  33.9148526 129.1277211
625 1100 1311    3036    28.3576366  56.7152732  94.9270903
529 1035 1496    3060    11.9687456  23.9374911 144.0937633

DECLARE FUNCTION gcd# (x#, y#)
DEFDBL A-Z
CLS
pi = 4 * ATN(1#)
FOR p = 1 TO 9999
  FOR s1 = 1 TO p / 3
    FOR s2 = s1 + 1 TO (p - s1) / 2
      s3 = p - s1 - s2
      g = gcd(s1, gcd(s2, s3))
      IF g = 1 AND s3 < s1 + s2 THEN
        c2a = (s1 * s1 + s3 * s3 - s2 * s2) / (2 * s1 * s3)
        ca = (s2 * s2 + s3 * s3 - s1 * s1) / (2 * s2 * s3)
        s2a = SQR(1 - c2a * c2a): sa = SQR(1 - ca * ca)
        IF c2a > 0 AND ca > 0 AND sa > 0 THEN
          twoA = ATN(s2a / c2a)
          a = ATN(sa / ca)
          ratio = twoA / a
          obt = pi - twoA - a
          IF obt + .000000001# > pi / 2 AND ABS(ratio - 2) < .000000001# THEN
             PRINT USING "#### #### ####    ####   ###.####### ###.####### ###.#######"; s1; s2; s3; p; a * 180 / pi; twoA * 180 / pi; (pi - twoA - a) * 180 / pi
          END IF
        END IF
      END IF
    NEXT
  NEXT
NEXT p

FUNCTION gcd (x, y)
 dnd = x: dvr = y
 IF dnd < dvr THEN SWAP dnd, dvr
 DO
   q = INT(dnd / dvr)
   r = dnd - q * dvr
   dnd = dvr: dvr = r
 LOOP UNTIL r = 0
 gcd = dnd
END FUNCTION


  Posted by Charlie on 2010-12-10 20:08:23
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