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%I #13 Jul 05 2023 03:00:16
%S 672,2688,6048,10752,15435,16800,19250,24192,32928,37730,43008,54432,
%T 56133,61740,67200,77000,81312,96768,113568,131712,138915,150920,
%U 151200,172032,173250,194208,217728,221130,224532,242592,246960,268800,296352,308000,325248,339570
%N Integer areas of the intangents triangle of integer-sided triangles.
%C Given triangle ABC, there are four lines simultaneously tangent to the incircle (with center I) and the excircle (with center J). Of these, three correspond to the sidelines of the triangle, and the fourth is known as the intangent (Kimberling 1998, p. 161), illustrated in the first link. The intangents intersect one another pairwise, and their points of intersection form the so-called intangents triangle A'B'C' illustrated in the second link.
%C The area of the intangents triangle is given by
%C S' = S*(a+b-c)^2*(a-b+c)^2*(-a+b+c)^2 /((-a^2+b^2+c^2)*(a^2+b^2-c^2)*(a^2-b^2+c^2)) where (a, b, c) are the sides of the initial triangle of ABC and S is the area of ABC.
%C Properties of this sequence:
%C The primitive triangles are 672, 15435, 19250, ...
%C The nonprimitive triangles of areas 4*a(n), 9*a(n), ..., p^2*a(n), ... are in the sequence.
%C The areas of the non-isosceles triangles are 15435, 19250, 37730, 56133, ...
%C The following table gives the first values (S', S, a, b, c, a', b', c') where S' is the area of the intangents triangle, S is the area of the triangles ABC, a, b, c the integer sides of the triangles ABC and a', b', c' are the integer sides of the intangents triangle.
%C ---------------------------------------
%C | S' | S | a | b | c |
%C ---------------------------------------
%C | 672 | 588 | 35 | 35 | 42 |
%C | 2688 | 2352 | 70 | 70 | 84 |
%C | 6048 | 5292 | 105 | 105 | 126 |
%C | 10752 | 9408 | 140 | 140 | 168 |
%C | 15435 | 12600 | 130 | 200 | 210 |
%C | 16800 | 14700 | 175 | 175 | 210 |
%C | 19250 | 9240 | 102 | 182 | 200 |
%C | 24192 | 21168 | 210 | 210 | 252 |
%C | 32928 | 28812 | 245 | 245 | 294 |
%C | 37730 | 36960 | 272 | 300 | 308 |
%C | 43008 | 37632 | 280 | 280 | 336 |
%C | 54432 | 47628 | 315 | 315 | 378 |
%C | 56133 | 44352 | 220 | 416 | 420 |
%D Clark Kimberling, Triangle centers and central triangles, Congressus Numerantium, 129 (1998)1-285.
%H Clark Kimberling, <a href="http://www.jstor.org/stable/2690608">Central points and central lines in the plane of a triangle</a>, Math. Magazine, 67(1994) 163-187.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IntangentsTriangle.html">Intangents Triangle</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IntangentsCircle.html">Intangents Circle</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Intangent.html">Intangent</a>.
%e 19250 is in the sequence from the initial triangle (102, 182, 200); we use the formula S' = S*(a+b-c)^2*(a-b+c)^2*(-a+b+c)^2 /((-a^2+b^2+c^2)*(a^2+b^2-c^2)*(a^2-b^2+c^2)) = 9240*(102+182-200)^2*(102-182+200)^2*(-102+182+200)^2 /((-102^2+182^2+200^2)*(102^2+182^2-200^2)*(102^2-182^2+200^2)) = 19250 where the area S = 9240 is obtained by Heron's formula S = sqrt(s*(s-a)*(s-b)*(s-c)) = sqrt(242*(242-102)*(242-182)*(242-200)) = 9240 with the semiperimeter s = (a+b+c)/2 = (102+182+200)/2 = 242.
%t nn = 1000; lst = {}; Do[s = (a + b + c)/2; n = (a + b - c)^2 *(a -b + c)^2 *(-a + b + c)^2;d =(-a^2 + b^2 + c^2) *(a^2 + b^2 - c^2) *(a^2 - b^2 + c^2); If[IntegerQ[s], area2 = s (s - a) (s - b) (s - c); If[area2>0 && d>0 &&IntegerQ[Sqrt[area2] * n/d], AppendTo[lst, Sqrt[area2] * n/d]]], {a, nn}, {b, a}, {c, b}]; Union[lst]
%Y Cf. A188158.
%K nonn
%O 1,1
%A _Michel Lagneau_, Nov 13 2013