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A231739 Integer areas of the Lucas Central triangles of integer-sided triangles. 1
2775, 11100, 24975, 34125, 44400, 69375, 99900, 135975, 136500, 177600, 224775, 277500, 307125, 335775, 399600, 468975, 543900, 546000, 624375, 710400, 801975, 853125, 899100, 1001775, 1110000, 1223775, 1228500, 1343100, 1467975, 1598400, 1672125, 1734375 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Consider a reference triangle ABC and externally inscribe a square on the side BC. Now join the new vertices S_AB and S_AC of this square with the vertex A, marking the points of intersection Q_ABC and Q_ACB. Next, draw lines perpendicular to the side BC through each of Q_ABC and Q_ACB. These points cross the sides AB and AC at Q_AB and Q_AC, respectively, resulting in an inscribed square Q_ABC Q_ACB Q_AB Q_AC. The circumcircle through A, Q_AB, and Q_AC is then known as the Lucas A-circles (Panakis 1973, p. 458; Yiu and Hatzipolakis 2001), and repeating the process for other sides gives the corresponding B- and C-circles.

The Lucas central triangle is the triangle L1 L2 L3 formed by the centers of the Lucas circles of a given reference triangle ABC.

The Lucas central triangle has side lengths

a' = 2*R*(a*b*c+b^2*R+c^2*R)*a/((a*c+2*b*R)*(a*b+2*c*R));

b' = 2*R*(a*b*c+a^2*R+c^2*R)*b/((b*c+2*a*R)*(a*b+2*c*R));

c' = 2*R*(a*b*c+a^2*R+b^2*R)*c/((b*c+2*a*R)*(a*c+2*b*R)).

Its area is given by

S' = a*b*c*R^2*sqrt(u)/v where:

u=3*a^2*b^2*c^2+4*a*b*c*(a^2+b^2+c^2)*R+4*(a^2*b^2+a^2*c^2+b^2*c^2)*R^2 ;

v=(b*c+2*a*R)*(a*c+2*b*R)*(a*b+2*c*R);

R = a*b*c/ sqrt((a+b+c)*(b+c-a)*(c+a-b)*(a+b-c)) = a*b*c/(4*S) where R is the circumradius of the reference triangle ABC and S its area.

Properties of this sequence:

The side lengths of the Lucas central triangles are rational numbers, sometimes integers, for example a(n) = 136500,...

The primitive Lucas central triangles are 2775, 34125,...

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

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

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

|   S'   |      S  |   a  |   b  |   c  |    a'    |    b'   |   c'    |

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

|   2775 |    8214 |  111 |  148 |  185 |  975/14  |  580/7  |  185/2  |

|  11100 |   32856 |  222 |  296 |  370 |  975/7   |  1160/7 |  185    |

|  24975 |   73926 |  333 |  444 |  555 |  2925/14 |  1740/7 |  555/2  |

|  34125 |  115248 |  490 |  490 |  588 |  545/2   |   545/2 |  300    |

|  44400 |  131424 |  444 |  592 |  740 |  1950/7  |  2320/7 |  370    |

|  69375 |  205350 |  555 |  740 |  925 |  4875/14 |  2900/7 |  925/2  |

|  99900 |  295704 |  666 |  888 | 1110 |  2925/7  |  3480/7 |  555    |

| 135975 |  402486 |  777 | 1036 | 1295 |   975/2  |   580   | 1295/2  |

| 136500 |  460992 |  980 |  980 | 1176 |   545    |   545   |  600    |

| 177600 |  525696 |  888 | 1184 | 1480 |  3900/7  |  4640/7 |  740    |

| 224775 |  665334 |  999 | 1332 | 1665 |  8775/14 |  5220/7 | 1665/2  |

| 277500 |  821400 | 1110 | 1480 | 1850 | 48755/7  |  5800/7 |  925    |

| 307125 | 1037232 | 1470 | 1470 | 1764 |  1635/2  |  1635/2 |  900    |

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

LINKS

Table of n, a(n) for n=1..32.

P. Moses, Circles and Triangle Centers Associated with the Lucas Circles, Forum Geom. 5, 97-106, 2005.

Wolfram MathWorld, Lucas Central Triangle

Wolfram MathWorld, Lucas Circles

P. Yiu and A. P. Hatzipolakis, The Lucas Circles of a Triangle, Amer. Math. Monthly 108, 444-446, 2001.

EXAMPLE

2775 is in the sequence. We use two ways:

First way: from the initial triangle (111, 148, 185) the formula in the comments gives directly the area of the Lucas central triangle: S' = a*b*c*R^2*sqrt(u)/v where:

R = a*b*c/4S = a*b*c/(4*sqrt(s(s-a)(s-b)(s-c)) = 111*148*185/4*sqrt(222(222-111)(222-148)(222-185)) = 111*148*185/(4*8214) = 185/2 with S=8214.

sqrt(u) = sqrt(3*a^2*b^2*c^2+4*a*b*c*(a^2+b^2+c^2)*R+4*(a^2*b^2+a^2*c^2+b^2*c^2)*R^2) = sqrt(154007727700225) = 12409985.

v = (b*c+2*a*R)*(a*c+2*b*R)*(a*b+2*c*R) = 116291549487925

And Sā€™ = 111*148*185*(185/2)^2*12409985/116291549487925 = 2775.

Second way: by calculation of the sides a', b', c' and by using Heron's formula. With the formulas given in the link, we find

aā€™ = 975/14;

bā€™ = 580/7;

cā€™ = 185/2.

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 = (975/14 + 580/7 + 185/2)/2 = 245/2;

We find S'= 2775.

MATHEMATICA

nn=2000; lst={}; Do[s =(a + b + c)/2; area2=s (s-a)(s-b)(s-c); If[area2>0, R = a*b*c /(4*Sqrt[area2]); u = a*b*c*R^2 * Sqrt[3*a^2*b^2*c^2 + 4*a*b*c*(a^2 + b^2 + c^2)*R + 4*(a^2*b^2 + a^2*c^2 + b^2*c^2)*R^2]; v = (b*c + 2*a*R)*(a*c + 2*b*R)*(a*b + 2*c*R); If[IntegerQ[u/v], AppendTo[lst, u/v]]], {a, nn}, {b, a}, {c, b}]; Union[lst]

CROSSREFS

Cf. A188158.

Sequence in context: A249228 A154081 A099691 * A251200 A188415 A236125

Adjacent sequences:  A231736 A231737 A231738 * A231740 A231741 A231742

KEYWORD

nonn

AUTHOR

Michel Lagneau, Nov 13 2013

STATUS

approved

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Last modified September 20 12:42 EDT 2017. Contains 292271 sequences.