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A231739 Integer areas of the Lucas Central triangles of integer-sided triangles.

%I A231739 #20 Feb 16 2025 08:33:20
%S A231739 2775,11100,24975,34125,44400,69375,99900,135975,136500,177600,224775,
%T A231739 277500,307125,335775,399600,468975,543900,546000,624375,710400,
%U A231739 801975,853125,899100,1001775,1110000,1223775,1228500,1343100,1467975,1598400,1672125,1734375
%N A231739 Integer areas of the Lucas Central triangles of integer-sided triangles.
%C A231739 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.
%C A231739 The Lucas central triangle is the triangle L1 L2 L3 formed by the centers of the Lucas circles of a given reference triangle ABC.
%C A231739 The Lucas central triangle has side lengths
%C A231739 a' = 2*R*(a*b*c+b^2*R+c^2*R)*a/((a*c+2*b*R)*(a*b+2*c*R));
%C A231739 b' = 2*R*(a*b*c+a^2*R+c^2*R)*b/((b*c+2*a*R)*(a*b+2*c*R));
%C A231739 c' = 2*R*(a*b*c+a^2*R+b^2*R)*c/((b*c+2*a*R)*(a*c+2*b*R)).
%C A231739 Its area is given by
%C A231739 S' = a*b*c*R^2*sqrt(u)/v where:
%C A231739 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 ;
%C A231739 v=(b*c+2*a*R)*(a*c+2*b*R)*(a*b+2*c*R);
%C A231739 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.
%C A231739 Properties of this sequence:
%C A231739 The side lengths of the Lucas central triangles are rational numbers, sometimes integers, for example a(n) = 136500, ...
%C A231739 The primitive Lucas central triangles are 2775, 34125, ...
%C A231739 The non-primitive triangles of areas 4*a(n), 9*a(n), ..., p^2*a(n), ... are in the sequence.
%C A231739 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.
%C A231739 +--------+---------+------+------+------+----------+---------+---------+
%C A231739 |    S'  |     S   |   a  |   b  |   c  |    a'    |    b'   |   c'    |
%C A231739 +--------+---------+------+------+------+----------+---------+---------+
%C A231739 |   2775 |    8214 |  111 |  148 |  185 |  975/14  |  580/7  |  185/2  |
%C A231739 |  11100 |   32856 |  222 |  296 |  370 |  975/7   |  1160/7 |  185    |
%C A231739 |  24975 |   73926 |  333 |  444 |  555 |  2925/14 |  1740/7 |  555/2  |
%C A231739 |  34125 |  115248 |  490 |  490 |  588 |  545/2   |   545/2 |  300    |
%C A231739 |  44400 |  131424 |  444 |  592 |  740 |  1950/7  |  2320/7 |  370    |
%C A231739 |  69375 |  205350 |  555 |  740 |  925 |  4875/14 |  2900/7 |  925/2  |
%C A231739 |  99900 |  295704 |  666 |  888 | 1110 |  2925/7  |  3480/7 |  555    |
%C A231739 | 135975 |  402486 |  777 | 1036 | 1295 |   975/2  |   580   | 1295/2  |
%C A231739 | 136500 |  460992 |  980 |  980 | 1176 |   545    |   545   |  600    |
%C A231739 | 177600 |  525696 |  888 | 1184 | 1480 |  3900/7  |  4640/7 |  740    |
%C A231739 | 224775 |  665334 |  999 | 1332 | 1665 |  8775/14 |  5220/7 | 1665/2  |
%C A231739 | 277500 |  821400 | 1110 | 1480 | 1850 | 48755/7  |  5800/7 |  925    |
%C A231739 | 307125 | 1037232 | 1470 | 1470 | 1764 |  1635/2  |  1635/2 |  900    |
%C A231739 +--------+---------+------+------+------+----------+---------+---------+
%H A231739 Peter J. C. Moses, <a href="http://forumgeom.fau.edu/FG2005volume5/FG200513index.html">Circles and Triangle Centers Associated with the Lucas Circles</a>, Forum Geom. 5, 97-106, 2005.
%H A231739 Wolfram MathWorld, <a href="https://mathworld.wolfram.com/LucasCentralTriangle.html">Lucas Central Triangle</a>
%H A231739 Wolfram MathWorld, <a href="https://mathworld.wolfram.com/LucasCircles.html">Lucas Circles</a>
%H A231739 P. Yiu and A. P. Hatzipolakis, <a href="http://www.jstor.org/stable/2695801">The Lucas Circles of a Triangle</a>, Amer. Math. Monthly 108, 444-446, 2001.
%e A231739 2775 is in the sequence. We use two ways:
%e A231739 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:
%e A231739 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.
%e A231739 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.
%e A231739 v = (b*c+2*a*R)*(a*c+2*b*R)*(a*b+2*c*R) = 116291549487925
%e A231739 And S' = 111*148*185*(185/2)^2*12409985/116291549487925 = 2775.
%e A231739 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
%e A231739 a' = 975/14;
%e A231739 b' = 580/7;
%e A231739 c' = 185/2.
%e A231739 Now, we use Heron's formula with (a', b', c'). We find A' = sqrt(s1*(s1-a')*(s1-b')*(s1-c')) with:
%e A231739 s1 =(a'+b'+c')/2 = (975/14 + 580/7 + 185/2)/2 = 245/2;
%e A231739 We find S'= 2775.
%t A231739 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]
%Y A231739 Cf. A188158.
%K A231739 nonn
%O A231739 1,1
%A A231739 _Michel Lagneau_, Nov 13 2013