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A230558 Integer areas of extouch triangles of integer-sided triangles.

%I A230558 #25 Feb 16 2025 08:33:20
%S A230558 30,48,72,84,120,192,252,270,288,336,432,480,648,720,750,756,768,780,
%T A230558 936,1008,1080,1152,1200,1344,1470,1728,1800,1920,2100,2268,2352,2400,
%U A230558 2430,2592,2784,2880,3000,3024,3060,3072,3120,3528,3600,3630,3888,4032,4116
%N A230558 Integer areas of extouch triangles of integer-sided triangles.
%C A230558 The extouch triangle T1T2T3 is the triangle formed by the points of tangency of a triangle ABC with its excircles J1, J2 and J3. The points T1, T2, and T3 can also be constructed as the points that bisect the perimeter of the triangle ABC starting at A, B, and C.
%C A230558 The side lengths of the extouch triangle are:
%C A230558 a'= sqrt(a^2 - 4*A^2/b*c)
%C A230558 b'= sqrt(b^2 - 4*A^2/a*c)
%C A230558 c'= sqrt(c^2 - 4*A^2/a*b)
%C A230558 where A is the triangle area of the original triangle.
%C A230558 The extouch triangle has area:
%C A230558 A*(a+b-c)*(a-b+c)*(-a+b+c)/4abc = A*2*r^2*s/(a*b*c) where r and s are the inradius and semiperimeter, respectively.
%C A230558 It is interesting to note that the sides of the extouch triangles are irrational numbers (in the general case) but the areas are integers.
%C A230558 The following table gives the first values (A', A, a, b, c,t1,t2,t3) where A' is the area of the extouch triangles, A is the area of the triangles ABC, a, b, c the integer sides of the original triangles ABC and t1, t2, t3 are the integer sides of the extouch triangles.
%C A230558 -------------------------------------------------------------
%C A230558 A'  |   A  | a | b  | c  |     t1    |     t2    |  t3
%C A230558 -------------------------------------------------------------
%C A230558 30  |  150 | 15| 20 | 25 | 3*sqrt(5) | 4*sqrt(10)|5*sqrt(13)
%C A230558 48  |  300 | 25| 25 | 40 | sqrt(265) | sqrt(265) | 32
%C A230558 72  |  300 | 25| 25 | 30 | sqrt(145) | sqrt(145) | 18
%C A230558 84  | 1050 | 35| 75 |100 | 7*sqrt(13)|3*sqrt(385)|8*sqrt(130)
%C A230558 120 |  600 | 30| 40 | 50 | 6*sqrt(5) |8*sqrt(10) |10*sqrt(13)
%C A230558 192 | 1200 | 50| 50 | 80 |2*sqrt(265)|2*sqrt(265)| 64
%C A230558 252 | 2100 | 35|120 |125 |    7      |72*sqrt(2) |5*sqrt(457)
%C A230558 270 | 1350 | 45| 60 | 75 |9*sqrt(5)  |12*sqrt(10)|15*sqrt(13)
%C A230558 288 | 1200 | 50| 50 | 60 |2*sqrt(145)|2*sqrt(145)| 36
%C A230558 336 | 4200 | 70|150 |200 |14*sqrt(13)|6*sqrt(485)|16*sqrt(130)
%C A230558 432 | 2700 | 75| 75 |120 |3*sqrt(265)|3*sqrt(265)| 96
%C A230558 480 | 2400 | 60| 80 |100 |12*sqrt(5) |16*sqrt(10)|20*sqrt(13)
%C A230558 648 | 2700 | 75| 75 | 90 |3*sqrt(145)|3*sqrt(145)| 54
%C A230558 ..................................................
%C A230558 Observation: the three altitudes of a majority of initial triangles ABC are integers, except very rare triangles, for example the initial triangle (35, 120, 125) where A = 2100 (see the following table).
%C A230558 This table gives the first values (A',A, h1, h2, h3) where A' is the area of the extouch triangles, A is the area of the initial triangles ABC and h1, h2, h3 are the altitudes of the initial triangles.
%C A230558 -------------------------------
%C A230558 A'  |  A   |  h1 |  h2 |  h3
%C A230558 -------------------------------
%C A230558 30  |  150 |  20 |  15 |  12
%C A230558 48  |  300 |  24 |  24 |  15
%C A230558 72  |  300 |  24 |  24 |  20
%C A230558 84  | 1050 |  60 |  28 |  21
%C A230558 120 |  600 |  40 |  30 |  24
%C A230558 192 | 1200 |  48 |  48 |  30
%C A230558 252 | 2100 | 120 |  35 | 168/5
%C A230558 270 | 1350 |  60 |  45 |  36
%C A230558 288 | 1200 |  48 |  48 |  40
%C A230558 336 | 4200 | 120 |  56 |  42
%C A230558 432 | 2700 |  72 |  72 |  45
%C A230558 480 | 2400 |  80 |  60 |  48
%C A230558 648 | 2700 |  72 |  72 |  60
%C A230558 ...............................
%H A230558 T. Dosa, <a href="http://forumgeom.fau.edu/FG2007volume7/FG200721index.html">Some Triangle Centers Associated with the Excircles</a>, Forum Geometricorum, Volume 7 (2007) 151-158.
%H A230558 C. Kimberling, <a href="http://faculty.evansville.edu/ck6/tcenters/tcct.html">Triangle Centers and Central Triangles</a>, Congr. Numer. 129, 1-295, 1998.
%H A230558 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ExtouchTriangle.html">Extouch Triangles</a>
%e A230558 30 is in the sequence. We use two ways:
%e A230558 First way: the formula A' = A*(a+b-c)*(a-b+c)*(-a+b+c)/(4*a*b*c) gives directly the result: A' = 150*(15+20-25)*(15-20+25)*(-15+20+25)/(4*15*20*25) = 30, with the area A = 150 obtained by Heron's formula A = sqrt(s*(s-a)*(s-b)*(s-c)) = sqrt(30*(30-15)*(30-20)*(30-25)) = 150, where s is the semiperimeter.
%e A230558 Second way: by calculation of the sides t1, t2, t3 and by using Heron's formula.
%e A230558 The extouch triangle (t1,t2,t3) of the initial triangle (a, b, c) = (15, 20, 25) is the triangle (3*sqrt(5), 4*sqrt(10), 5*sqrt(13)) where:
%e A230558 a' = sqrt(a^2 - 4*A^2/b*c) = sqrt(15^2-4*150^2/(20*25)) = 3*sqrt(5);
%e A230558 b' = sqrt(b^2 - 4*A^2/a*c) = sqrt(20^2-4*150^2/(15*25)) = 4*sqrt(10);
%e A230558 c' = sqrt(c^2 - 4*A^2/a*b) = sqrt(25^2 - 4*150^2/(15*20)) = 5*sqrt(13).
%e A230558 Now, we use Heron's formula with (t1,t2,t3). We find A'=sqrt(s1*(s1-t1)*(s1-t2)*(s1-t3))with:
%e A230558 s1 =(t1+t2+t3)/2 = (3*sqrt(5)+ 4*sqrt(10) + 5*sqrt(13))/2;
%e A230558 We find A'= 30.
%t A230558 nn = 1000; lst = {}; Do[s = (a + b + c)/2; If[IntegerQ[s], area2 = s (s - a) (s - b) (s - c); t = Sqrt[area2]*(a + b - c)*(a - b + c)*(-a + b + c)/(4*a*b*c); If[0 < area2 && IntegerQ[Sqrt[area2]] && IntegerQ[t], AppendTo[lst, t]]], {a, nn}, {b, a}, {c, b}]; Union[lst]
%Y A230558 Cf. A188158, A210643.
%K A230558 nonn
%O A230558 1,1
%A A230558 _Michel Lagneau_, Oct 23 2013