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A231328 Integer areas of the reflection triangles of integer-sided triangles. 0
18, 72, 90, 162, 180, 252, 288, 360, 450, 540, 630, 648, 720, 810, 882, 990, 1008, 1152, 1440, 1458, 1512, 1620, 1638, 1800, 1890, 2160, 2178, 2250, 2268, 2520, 2592, 2772, 2880, 2970, 3042, 3240, 3528, 3672, 3960, 4032, 4050, 4158, 4410, 4500, 4608, 4680, 4860 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The triangle A'B'C' obtained by reflecting the vertices of a reference triangle ABC about the opposite sides is called the reflection triangle (Grinberg 2003).

The area of the reflection triangle is given by

A' = A*t/(a^2*b^2*c^2) where A is the area of the reference triangle of sides (a, b, c) and

t=-(a^6-b^2*a^4-c^2*a^4-b^4*a^2-c^4*a^2-b^2*c^2*a^2+b^6+c^6-b^2*c^4-b^4*c^2)/(a^2*b^2*c^2).

See the link for the side lengths of the reflection triangles.

Properties of this sequence:

The areas corresponding to the primitive reflection triangles are 18, 90, 180, 252, 540,...

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

It appears that one of the side of the reflection triangles equals the greatest side of the initial triangle (see the table below), and the initial triangles are Pythagorean triangles => a(n) = 3*A009112(n).

The following table gives the first values (A, A', a, b, c, a', b', c') where A' is the area of the reflection triangles, A is the area of the initial triangles, a, b, c are the integer sides of the initial triangles, and a', b', c' are the sides of the reflection triangles.

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

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

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

|  18  |   6 |  3 |  4 |   5 |  9*sqrt(17)/5    |  4*sqrt(97)/5    |  5 |

|  72  |  24 |  6 |  8 |  10 | 18*sqrt(17)/5    |  8*sqrt(97)/5    | 10 |

|  90  |  30 |  5 | 12 |  13 |  5*sqrt(1321)/13 | 36*sqrt(41)/13   | 13 |

| 162  |  54 |  9 | 12 |  15 | 27*sqrt(17)/5    | 12*sqrt(97)/5    | 15 |

| 180  |  60 |  8 | 15 |  17 |  8*sqrt(2089)/17 | 45*sqrt(89)/17   | 17 |

| 252  |  84 |  7 | 24 |  25 |  7*sqrt(5233)/25 | 72*sqrt(113)/25  | 25 |

| 288  |  96 | 12 | 16 |  20 | 36*sqrt(17)/5    | 16*sqrt(97)/5    | 20 |

| 360  | 120 | 10 | 24 |  26 | 10*sqrt(1321)/13 | 72*sqrt(41)/13   | 26 |

| 450  | 150 | 15 | 20 |  25 |  9*sqrt(17)      |  4*sqrt(97)      | 25 |

| 540  | 180 |  9 | 40 |  41 | 27*sqrt(1609)/41 | 40*sqrt(2329)/41 | 41 |

| 630  | 210 | 12 | 35 |  37 | 36*sqrt(1241)/37 | 35*sqrt(2521)/37 | 37 |

| 648  | 216 | 18 | 24 |  30 | 54*sqrt(17)/5    | 24*sqrt(97)/5    | 30 |

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

REFERENCES

D. Grinberg, On the Kosnita Point and the Reflection Triangle, Forum Geom. 3, 105-111, 2003.

LINKS

Table of n, a(n) for n=1..47.

Eric W. Weisstein, MathWorld: Reflection Triangle

EXAMPLE

18 is in the sequence. We use two ways:

First way: with the triangle (3, 4, 5) the formula A' = A*t/(a^2*b^2*c^2) gives directly the result: A'= 18 where the area A = 6 is obtained by Heron's formula A =sqrt(s*(s-a)*(s-b)*(s-c))= sqrt(6*(6-3)*(6-4)*(6-5)) = 6, where s is the semiperimeter.

Second way: by calculation of the sides a', b', c' and by using Heron's formula. We obtain from the formulas given in the link:

a' = 9*sqrt(17)/5;

b' = 4*sqrt(97/5);

c' = 5.

Now, we use Heron's formula with (a',b',c'). We find A'=sqrt(s1*(s1-a')*(s1-b')*(s1-c')) with:

s1 =(a'+b'+c')/2 = (9*sqrt(17)/5+ 4*sqrt(97/5)+ 5)/2. We find A'= 18.

MATHEMATICA

nn = 300 ; lst = {}; Do[s = (a + b + c)/2 ; If[IntegerQ[s], area2 = s (s-a)(s-b) (s-c); If[area2 > 0 && IntegerQ[Sqrt[area2] + (a^2 + b^2 + c^2)/8], AppendTo[lst, Sqrt[area2] + (a^2 + b^2 + c^2)/8]]], {a, nn}, {b, a}, {c, b}] ; Union[lst]

CROSSREFS

Cf. A188158, A009111, A009112.

Sequence in context: A174492 A088490 A257693 * A274577 A195321 A069058

Adjacent sequences:  A231325 A231326 A231327 * A231329 A231330 A231331

KEYWORD

nonn

AUTHOR

Michel Lagneau, Nov 07 2013

STATUS

approved

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Last modified November 17 13:32 EST 2019. Contains 329230 sequences. (Running on oeis4.)