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Very Large Triangles (Posted on 2019-11-21) Difficulty: 4 of 5
How many different triangles with integer side lengths are there such that the sum of the lengths of any 2 sides is at least 5 units more than the length of the third side, and that the area is numerically twice the perimeter?

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution Solution | Comment 1 of 2
I found 8

 x  y  z area perim
 9 40 41 180  90
 9 75 78 324 162
10 24 26 120  60
10 35 39 168  84
11 25 30 132  66
12 16 20  96  48
13 14 15  84  42
15 15 24 108  54

I haven't proved that there are none for triangles with a side larger than 1,000.

Code follows:

count = 0
threshold = .0001
print('  x  ' , 'y   ' , 'z ' , 'diff,' , 'area,' ,  'perimeter')
for x in (range(1,1000)):
    for y in (range(x,1000)):
        for z in (range(y,1000)):
            if x+y-z < 5:
                continue

            s = (x+y+z)/2
            area = ((s*(s-x)*(s-y)*(s-z))**.5)

            if abs(area - 2*(x+y+z)) < threshold:
                    print(x , y,  z,  int(area), x+y+z)
                    count += 1
           
print('number of solutions is: ', count)


Edited on November 21, 2019, 11:26 am
  Posted by Larry on 2019-11-21 11:14:56

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