SOLUTION: There are how many triangles with perimeter 20 and integer sides such that altitudes ratio is 1:2:3

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Question 460423: There are how many triangles with perimeter 20 and integer sides such that altitudes ratio is 1:2:3
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
None.
Let a, b, and c be the lengths of the sides of the triangle.
Then by the given, a + b + c = 20.
Now the altitudes being in the ratio 1:2:3 means that if x is the shortest altitude , then the longer altitude is 2x, and the longest altitude is 3x.
Without loss of generality, let x be the altitude for a, 2x be the altitude for b, and 3x be the altitude for c.
Then by using the area formula, we get
%28ax%29%2F2+=+%282bx%29%2F2+=+%283cx%29%2F2 <==> a = 2b = 3c.
==> c = a/3 and b = a/2.
==> a + a/2 + a/3 = 20
==> %2811a%29%2F6+=+20 ==> a+=+120%2F11.
Since a is not an integer, neither will b and c be.