login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A230758 Integer areas of the first Neuberg triangles of integer-sided triangles. 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

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

LINKS

Table of n, a(n) for n=1..35.

Wolfram MathWorld, First Neuberg Triangles

EXAMPLE

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.

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]

CROSSREFS

Cf. A188158.

Sequence in context: A282328 A187465 A325604 * A318201 A236013 A282440

Adjacent sequences:  A230755 A230756 A230757 * A230759 A230760 A230761

KEYWORD

nonn

AUTHOR

Michel Lagneau, Oct 29 2013

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 21 02:29 EDT 2021. Contains 348141 sequences. (Running on oeis4.)