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A230758 Integer areas of the first Neuberg triangles of integer-sided triangles.

%I A230758 #16 Feb 16 2025 08:33:20
%S A230758 1275,4614,5100,7275,11475,18456,20400,29100,31875,41526,45900,60409,
%T A230758 62475,65475,73824,81600,103275,103935,115350,116400,127500,154275,
%U A230758 166104,173229,181875,183600,215475,226086,241636,249900,261900,265974,286875,295296,434209
%N A230758 Integer areas of the first Neuberg triangles of integer-sided triangles.
%C A230758 The triangle LMN formed by joining a set of three Neuberg centers (i.e., centers of the Neuberg circles) is obtained from the edges of a given triangle ABC (see the figure given in the link). Similarly, three reflected Neuberg circles with centers L', M', and N' can be obtained from the main circles by reflection in their respective sides of the triangle, producing a reflected Neuberg triangle L'M'N'.
%C A230758 The Neuberg triangle has area
%C A230758 A' = (a^2*b^2 + a^2*c^2 + b^2*c^2)/(4*sqrt((-a+b+c)*(a-b+c)*(a+b-c)*(a+b+c))).
%C A230758 And the side lengths are
%C A230758 a' = sqrt((b^6 -a^2*b^2*c^2 + b^4*c^2 + b^2*c^4 + c^6)/d)
%C A230758 b' = sqrt((a^6 -a^2*b^2*c^2 + a^4*c^2 + a^2*c^4 + c^6)/d)
%C A230758 c' = sqrt((a^6 -a^2*b^2*c^2 + a^4*b^2 + a^2*b^4 + b^6)/d)
%C A230758 with d = (-a+b+c)*(a-b+c)*(a+b-c)*(a+b+c).
%C A230758 Properties of this sequence:
%C A230758 The areas of the original triangles are integers. The primitive triangles with areas a(n) are 1275, 4614, 7275, 11475, ...
%C A230758 The non-primitive triangles with areas 4*a(n), 9*a(n), ..., p^2*a(n), ... are in the sequence.
%C A230758 It appears that if the original triangle is isosceles, the corresponding Neuberg triangle is also isosceles and the greatest side is an integer divisible by 25.
%C A230758 It is interesting to note that the sides of the Neuberg triangle are irrational numbers (in the general case) and the areas are integers.
%C A230758 The following table gives the first values (A',A,a,b,c,a',b',c') where A' is the area of the Neuberg triangles, A is the area of the reference triangles ABC, a, b, c the integer sides of the original triangles ABC and a', b', c' the integer sides of the Neuberg triangles.
%C A230758 ------------------------------------------------------------------------
%C A230758 |   A'|   A |  a |  b |  c |       a'      |       b'      |   c'
%C A230758 ------------------------------------------------------------------------
%C A230758 | 1275|  768|  40|  40|  64|  sqrt(42241)/2|  sqrt(42241)/2|   25
%C A230758 | 4614| 3456|  72|  96| 120|  sqrt(32521)  |2*sqrt(5101)   |5*sqrt(193)
%C A230758 | 5100| 3072|  80|  80| 128|  sqrt(42241)  |  sqrt(42241)  |   50
%C A230758 | 7275| 6912| 120| 120| 144|  sqrt(94681)/2|  sqrt(94681)/2|  100
%C A230758 |11475| 6912| 120| 120| 192|3*sqrt(42241)/2|3*sqrt(42241)/2|   75
%C A230758 |18456|13824| 144| 192| 240|2*sqrt(32521)  |4*sqrt(5101)   |10*sqrt(193)
%C A230758 |20400|12288| 160| 160| 256|2*sqrt(42241)  |2*sqrt(42241)  |  100
%C A230758 |29100|27648| 240| 240| 288|  sqrt(94681)  |  sqrt(94681)  |  200
%C A230758 |31875|19200| 200| 200| 320|5*sqrt(42241)/2|5*sqrt(42241)/2|  125
%C A230758 |41526|31104| 216| 288| 360|3*sqrt(32521)  |6*sqrt(5101)   |15*sqrt(193)
%C A230758 |45900|27648| 240| 240| 384|3*sqrt(42241)  |3*sqrt(42241)  |  150
%C A230758 ......................................................
%H A230758 Wolfram MathWorld, <a href="https://mathworld.wolfram.com/FirstNeubergTriangle.html">First Neuberg Triangles</a>
%e A230758 4614 is in the sequence because the area A' = (a^2*b^2 + a^2*c^2 + b^2*c^2)/(4*sqrt((-a+b+c)*(a-b+c)*(a+b-c)*(a+b+c))) of the first Neuberg triangle corresponding to the initial triangle (72,96,120) is A' = (72^2*96^2 + 72^2*120^2 + 96^2*120^2)/(4*sqrt((-72+96+120)*(72-96+120)*(72+96-120)*(72+96+120))) = 4614.
%t A230758 nn = 1000; lst = {}; Do[d=(-a+b+c)(a-b+c)(a+b-c)(a+b+c); If[d>0 && IntegerQ[(a^2*b^2+a^2*c^2+b^2*c^2)/(4*Sqrt[d])], AppendTo[lst, (a^2*b^2+a^2*c^2+b^2*c^2)/(4*Sqrt[d])]], {a, nn}, {b, a}, {c, b}]; Union[lst]
%Y A230758 Cf. A188158.
%K A230758 nonn
%O A230758 1,1
%A A230758 _Michel Lagneau_, Oct 29 2013