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.

DEFDBL A-Z
CLS
pi = 4 * ATN(1#)
FOR p = 1 TO 999
  FOR s1 = 1 TO p / 3
    FOR s2 = s1 + 1 TO (p - s1) / 2
      s3 = p - s1 - s2
      IF 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

finds

   sides       peri        angles (degrees)
 16  28  33      77    28.9550244  57.9100487  93.1349269
 25  45  56     126    25.8419328  51.6838655 102.4742017
 32  56  66     154    28.9550244  57.9100487  93.1349269
 36  66  85     187    23.5564643  47.1129286 109.3306071
 48  84  99     231    28.9550244  57.9100487  93.1349269
 50  90 112     252    25.8419328  51.6838655 102.4742017
 49  91 120     260    21.7867893  43.5735786 114.6396321
 64 112 132     308    28.9550244  57.9100487  93.1349269
 64 120 161     345    20.3641348  40.7282696 118.9075956
 72 132 170     374    23.5564643  47.1129286 109.3306071
 75 135 168     378    25.8419328  51.6838655 102.4742017
 80 140 165     385    28.9550244  57.9100487  93.1349269
 81 144 175     400    27.2660445  54.5320889  98.2018666
 81 153 208     442    19.1881365  38.3762729 122.4355906
 96 168 198     462    28.9550244  57.9100487  93.1349269
100 180 224     504    25.8419328  51.6838655 102.4742017
 98 182 240     520    21.7867893  43.5735786 114.6396321
112 196 231     539    28.9550244  57.9100487  93.1349269
100 190 261     551    18.1948723  36.3897447 125.4153830
108 198 255     561    23.5564643  47.1129286 109.3306071
128 224 264     616    28.9550244  57.9100487  93.1349269
121 220 279     620    24.6199773  49.2399547 106.1400680
125 225 280     630    25.8419328  51.6838655 102.4742017
121 231 320     672    17.3414428  34.6828856 127.9756716
128 240 322     690    20.3641348  40.7282696 118.9075956
144 252 297     693    28.9550244  57.9100487  93.1349269
144 264 340     748    23.5564643  47.1129286 109.3306071
150 270 336     756    25.8419328  51.6838655 102.4742017
160 280 330     770    28.9550244  57.9100487  93.1349269
147 273 360     780    21.7867893  43.5735786 114.6396321
162 288 350     800    27.2660445  54.5320889  98.2018666
144 276 385     805    16.5978421  33.1956843 130.2064736
169 299 360     828    27.7957725  55.5915450  96.6126825
176 308 363     847    28.9550244  57.9100487  93.1349269
175 315 392     882    25.8419328  51.6838655 102.4742017
162 306 416     884    19.1881365  38.3762729 122.4355906
169 312 407     888    22.6198649  45.2397299 112.1404052
192 336 396     924    28.9550244  57.9100487  93.1349269
180 330 425     935    23.5564643  47.1129286 109.3306071
169 325 456     950    15.9423686  31.8847372 132.1728942
196 350 429     975    26.7655006  53.5310012  99.7034983

as all the cases with perimeter under 1000, so the smallest perimeter is 77.

Edited on December 10, 2010, 2:06 pm

Posted by Charlie on 2010-12-10 14:05:11