cp's OEIS Frontend

A231740 Integer areas of the intangents triangle of integer-sided triangles.

%I A231740 #14 Feb 16 2025 08:33:20
%S A231740 672,2688,6048,10752,15435,16800,19250,24192,32928,37730,43008,54432,
%T A231740 56133,61740,67200,77000,81312,96768,113568,131712,138915,150920,
%U A231740 151200,172032,173250,194208,217728,221130,224532,242592,246960,268800,296352,308000,325248,339570
%N A231740 Integer areas of the intangents triangle of integer-sided triangles.
%C A231740 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 A231740 The area of the intangents triangle is given by
%C A231740 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 A231740 Properties of this sequence:
%C A231740 The primitive triangles are 672, 15435, 19250, ...
%C A231740 The nonprimitive triangles of areas 4*a(n), 9*a(n), ..., p^2*a(n), ... are in the sequence.
%C A231740 The areas of the non-isosceles triangles are 15435, 19250, 37730, 56133, ...
%C A231740 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 A231740 ---------------------------------------
%C A231740 |   S'  |     S  |   a  |   b   |  c  |
%C A231740 ---------------------------------------
%C A231740 |   672 |   588  |   35 |   35  |  42 |
%C A231740 |  2688 |  2352  |   70 |   70  |  84 |
%C A231740 |  6048 |  5292  |  105 |  105  | 126 |
%C A231740 | 10752 |  9408  |  140 |  140  | 168 |
%C A231740 | 15435 | 12600  |  130 |  200  | 210 |
%C A231740 | 16800 | 14700  |  175 |  175  | 210 |
%C A231740 | 19250 |  9240  |  102 |  182  | 200 |
%C A231740 | 24192 | 21168  |  210 |  210  | 252 |
%C A231740 | 32928 | 28812  |  245 |  245  | 294 |
%C A231740 | 37730 | 36960  |  272 |  300  | 308 |
%C A231740 | 43008 | 37632  |  280 |  280  | 336 |
%C A231740 | 54432 | 47628  |  315 |  315  | 378 |
%C A231740 | 56133 | 44352  |  220 |  416  | 420 |
%D A231740 Clark Kimberling, Triangle centers and central triangles, Congressus Numerantium, 129 (1998)1-285.
%H A231740 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 A231740 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IntangentsTriangle.html">Intangents Triangle</a>.
%H A231740 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IntangentsCircle.html">Intangents Circle</a>.
%H A231740 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Intangent.html">Intangent</a>.
%e A231740 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 A231740 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 A231740 Cf. A188158.
%K A231740 nonn
%O A231740 1,1
%A A231740 _Michel Lagneau_, Nov 13 2013