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

1275, 4614, 5100, 7275, 11475, 18456, 20400, 29100, 31875, 41526, 45900, 60409, 62475, 65475, 73824, 81600, 103275, 103935, 115350, 116400, 127500, 154275, 166104, 173229, 181875, 183600, 215475, 226086, 241636, 249900, 261900, 265974, 286875, 295296, 434209

Offset: 1

Author

Michel Lagneau, Oct 29 2013

Keywords

• nonn

Comments

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'.

The Neuberg triangle has 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))).

And the side lengths are

a' = sqrt((b^6 -a^2*b^2*c^2 + b^4*c^2 + b^2*c^4 + c^6)/d)

b' = sqrt((a^6 -a^2*b^2*c^2 + a^4*c^2 + a^2*c^4 + c^6)/d)

c' = sqrt((a^6 -a^2*b^2*c^2 + a^4*b^2 + a^2*b^4 + b^6)/d)

with d = (-a+b+c)*(a-b+c)*(a+b-c)*(a+b+c).

Properties of this sequence:

The areas of the original triangles are integers. The primitive triangles with areas a(n) are 1275, 4614, 7275, 11475, ...

The non-primitive triangles with areas 4*a(n), 9*a(n), ..., p^2*a(n), ... are in the sequence.

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.

It is interesting to note that the sides of the Neuberg triangle are irrational numbers (in the general case) and the areas are integers.

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.

------------------------------------------------------------------------

| A'| A | a | b | c | a' | b' | c'

------------------------------------------------------------------------

| 1275| 768| 40| 40| 64| sqrt(42241)/2| sqrt(42241)/2| 25

| 4614| 3456| 72| 96| 120| sqrt(32521) |2*sqrt(5101) |5*sqrt(193)

| 5100| 3072| 80| 80| 128| sqrt(42241) | sqrt(42241) | 50

| 7275| 6912| 120| 120| 144| sqrt(94681)/2| sqrt(94681)/2| 100

|11475| 6912| 120| 120| 192|3*sqrt(42241)/2|3*sqrt(42241)/2| 75

|18456|13824| 144| 192| 240|2*sqrt(32521) |4*sqrt(5101) |10*sqrt(193)

|20400|12288| 160| 160| 256|2*sqrt(42241) |2*sqrt(42241) | 100

|29100|27648| 240| 240| 288| sqrt(94681) | sqrt(94681) | 200

|31875|19200| 200| 200| 320|5*sqrt(42241)/2|5*sqrt(42241)/2| 125

|41526|31104| 216| 288| 360|3*sqrt(32521) |6*sqrt(5101) |15*sqrt(193)

|45900|27648| 240| 240| 384|3*sqrt(42241) |3*sqrt(42241) | 150

......................................................

Examples

			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.
		

Links

• Wolfram MathWorld, First Neuberg Triangles

Crossrefs

Cf. A188158.

Programs

Mathematica

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]