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A210643
Areas A of the triangles such that A, the sides and the three altitudes are integers.
6
150, 300, 600, 1050, 1200, 1350, 2400, 2700, 3750, 4200, 4800, 5070, 5400, 7350, 7500, 9450, 9600, 10140, 10800, 12150, 14700, 15000, 16800, 17340, 18150, 19200, 20280, 21600, 24300, 25350, 26250, 29400, 30000, 33750, 34680, 36300, 37800
OFFSET
1,1
COMMENTS
Properties of this sequence :
There exists three class of numbers included into a(n) :
(i) A subset such that {150, 600, 1350, 2400, 3750, 5070,…} where the sides a<b<c have the property a^2 + b^2 = c^2 => h1 = b, h2 = a, h3 = a*b/c.
(ii) A subset such that a(n) = 300*n^2 = {300, 1200, 2700, 4800, …} where the triangles (a,b,c) are isosceles with a = b < c, and it is easy to check that a = b = 25*n, c=30*n, h1 = h2 = 24*n and h3 = sqrt(b^2 - c^2/4).
(iii) A subset such that {1050, 4200, 9450,…} without the precedent properties.
LINKS
Eric Weisstein, Isosceles Triangle
Eric Weisstein, Altitude
EXAMPLE
Primitive solutions follow:
Area, ( a, b, c), (h1, h2, h3), Case
150, (15, 20, 25), (20, 15, 12), Right,
300, (25, 25, 30), (24, 24, 20), Isosceles,
300, (25, 25, 40), (24, 24, 15), Isosceles,
1050, (35, 75, 100), (60, 28, 21), Other,
5070, (65, 156, 169), (156, 65, 60), Right,
10140, (130, 169, 169), (156, 120, 120), Isosceles,
10140, (169, 169, 312), (120, 120, 65), Isosceles,
17340, (136, 255, 289), (255, 136, 120), Right,
34680, (272, 289, 289), (255, 240, 240), Isosceles,
34680, (289, 289, 510), (240, 240, 136), Isosceles,
52500, (175, 600, 625), (600, 175, 168), Right,
82500, (275, 625, 750), (600, 264, 220), Other.
MAPLE
with(numtheory):T:=array(1..1000):k:=0:nn:=500:for a from 1 to nn do: for b from a to nn do: for c from b to nn do:p:=(a+b+c)/2:s:=p*(p-a)*(p-b)*(p-c):if s>0 then s1:=sqrt(s): h1:=2*s1/a: h2:=2*s1/b:h3:=2*s1/c:if s1=floor(s1) and h1=floor(h1) and h2=floor(h2) and h3=floor(h3) then k:=k+1:T[k]:=s1:else fi:fi:od:od:od: L := [seq(T[i], i=1..k)]:L1:=convert(T, set):A:=sort(L1, `<`): print(A):
MATHEMATICA
nn = 900; lst = {}; Do[s = (a + b + c)/2; If[IntegerQ[s], area2 = s (s - a) (s - b) (s - c); If[0 < area2 <= nn^2 && IntegerQ[Sqrt[area2]]&&IntegerQ[(2*Sqrt[area2])/a] &&IntegerQ[(2*Sqrt[area2])/b] &&IntegerQ[(2*Sqrt[area2])/c], AppendTo[lst, Sqrt[area2]]]], {a, nn}, {b, a}, {c, b}]; Union[lst]
CROSSREFS
Cf. A188158.
Sequence in context: A335145 A063829 A291959 * A211550 A212465 A273322
KEYWORD
nonn
AUTHOR
Michel Lagneau, Mar 26 2012
EXTENSIONS
More terms from Ray Chandler, Apr 24 2013
STATUS
approved