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Tidy Triangle (Posted on 2006-05-05) Difficulty: 4 of 5
Find the smallest obtuse triangle such that its sides, the altitude to the obtuse angle, and the median to the largest side are all integers.

In case it arises, "smallest" refers to the area of the triangle.

See The Solution Submitted by Jer    
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Solution computer trial-and-error solution (spoiler) | Comment 1 of 7

Computer trial and error leads to a triangle with sides 25, 39 and 56, with a median of 17 and altitude of 15.

The display also shows the angle in degrees on the right side of the first line:

25  39  56                                    24.22775    9.10460  146.66765
17                                     15

There exist larger such triangles, but the following display, with two rows per triangle, shows also some spurious results, that have almost, but not quite, integral altitudes:

sides                                              angles (degrees)
median                                altitude
50  78  112                                   24.22775    9.10460  146.66765
33.99999999999999                      30
75  117  168                                  24.22775    9.10460  146.66765
50.99999999999999                      44.99999999999999
100  156  224                                 24.22775    9.10460  146.66765
67.99999999999999                      59.99999999999999
125  195  280                                 24.22775    9.10460  146.66765
84.99999999999999                      74.99999999999999
150  234  336                                 24.22775    9.10460  146.66765
102                                    89.99999999999999
139  327  370                                 58.11546    8.53232  113.35222
170                                    121.9999995210097
175  273  392                                 24.22775    9.10460  146.66765
119                                    105
212  238  390                                 18.19679   14.08910  147.71411
113                                    112
200  312  448                                 24.22775    9.10460  146.66765
136                                    120
105  445  500                                 46.84761    2.07802  131.07437
205                                    84
225  351  504                                 24.22775    9.10460  146.66765
153                                    135
250  390  560                                 24.22775    9.10460  146.66765
170                                    150
275  429  616                                 24.22775    9.10460  146.66765
187                                    165
300  468  672                                 24.22775    9.10460  146.66765
204                                    180
325  507  728                                 24.22775    9.10460  146.66765
221                                    195
278  654  740                                 58.11546    8.53232  113.35222
340                                    243.9999990420194
232  650  798                                 33.29749    3.57721  143.12531
281                                    160
350  546  784                                 24.22775    9.10460  146.66765
238                                    210
424  476  780                                 18.19679   14.08910  147.71411
226                                    224
375  585  840                                 24.22775    9.10460  146.66765
255                                    225
400  624  896                                 24.22775    9.10460  146.66765
271.9999999999999                      240
241  777  952                                 25.93495    2.14578  151.91927
323                                    148.999999763032

Many of the above are integral multiples of the smallest such triangle.  Those altitudes where the fractional part consists of all 9's are really integers and the amount off is due to rounding errors. But the ones where the last several places after the decimal are not 9 are really only close to integral in value.  I can theorize that since we are varying a total of 9 digits in the three side lengths, that enough triangles exist that probabilistically speaking, some would have altitudes close to, but not, integers.

The smallest triangles of a given shape (defining a set of similar triangles) are:

25  39  56                                    24.22775    9.10460  146.66765
17                                     15
212  238  390                                 18.19679   14.08910  147.71411
113                                    112
105  445  500                                 46.84761    2.07802  131.07437
205                                    84
232  650  798                                 33.29749    3.57721  143.12531
281                                    160

... in ascending order of perimeter.  By area, the middle two would be switched.

The list probably continues, but the trial and error takes more time as the sizes get larger.

DEFDBL A-Z
pi = ATN(1) * 4
CLS
FOR t = 4 TO 1000000
  FOR s1 = 1 TO INT(t / 3)
   FOR s2 = s1 + 1 TO INT((t - s1) / 2)
    s3 = t - s1 - s2
    IF s3 * s3 > s1 * s1 + s2 * s2 AND s3 < s1 + s2 THEN
     cosA = (s3 * s3 + s1 * s1 - s2 * s2) / (2 * s3 * s1)
     b = s3 / 2
     med = SQR(s1 * s1 + b * b - 2 * b * s1 * cosA)
     medi = INT(med + .5)
     IF ABS(med - medi) / med < .00000001# THEN
       alt = s1 * SQR(1 - cosA * cosA)
       alti = INT(alt + .5)
       IF ABS(alt - alti) / alt < .00000001# THEN
         cosC = (s3 * s3 + s2 * s2 - s1 * s1) / (2 * s3 * s2)
         A = ATN((1 - cosA * cosA) / cosA) * 180 / pi
         C = ATN((1 - cosC * cosC) / cosC) * 180 / pi

         PRINT s1; s2; s3; TAB(45);
         PRINT USING "  ###.#####"; A; C; 180 - A - C
         PRINT med; TAB(40); alt
         ct = ct + 1
         IF ct > 22 THEN END
       END IF
     END IF
    END IF
   NEXT
  NEXT
NEXT

so I did not have enough zeros in .00000001# to weed out the spurious values shown.


  Posted by Charlie on 2006-05-05 15:32:40
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