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 Minimum perimeter length (Posted on 2010-12-10)
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)

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 this program finds... (spoiler) | Comment 1 of 5

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  93.1349269100 180 224     504    25.8419328  51.6838655 102.4742017 98 182 240     520    21.7867893  43.5735786 114.6396321112 196 231     539    28.9550244  57.9100487  93.1349269100 190 261     551    18.1948723  36.3897447 125.4153830108 198 255     561    23.5564643  47.1129286 109.3306071128 224 264     616    28.9550244  57.9100487  93.1349269121 220 279     620    24.6199773  49.2399547 106.1400680125 225 280     630    25.8419328  51.6838655 102.4742017121 231 320     672    17.3414428  34.6828856 127.9756716128 240 322     690    20.3641348  40.7282696 118.9075956144 252 297     693    28.9550244  57.9100487  93.1349269144 264 340     748    23.5564643  47.1129286 109.3306071150 270 336     756    25.8419328  51.6838655 102.4742017160 280 330     770    28.9550244  57.9100487  93.1349269147 273 360     780    21.7867893  43.5735786 114.6396321162 288 350     800    27.2660445  54.5320889  98.2018666144 276 385     805    16.5978421  33.1956843 130.2064736169 299 360     828    27.7957725  55.5915450  96.6126825176 308 363     847    28.9550244  57.9100487  93.1349269175 315 392     882    25.8419328  51.6838655 102.4742017162 306 416     884    19.1881365  38.3762729 122.4355906169 312 407     888    22.6198649  45.2397299 112.1404052192 336 396     924    28.9550244  57.9100487  93.1349269180 330 425     935    23.5564643  47.1129286 109.3306071169 325 456     950    15.9423686  31.8847372 132.1728942196 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

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