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 A230558 Integer areas of extouch triangles of integer-sided triangles. 0
 30, 48, 72, 84, 120, 192, 252, 270, 288, 336, 432, 480, 648, 720, 750, 756, 768, 780, 936, 1008, 1080, 1152, 1200, 1344, 1470, 1728, 1800, 1920, 2100, 2268, 2352, 2400, 2430, 2592, 2784, 2880, 3000, 3024, 3060, 3072, 3120, 3528, 3600, 3630, 3888, 4032, 4116 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. The side lengths of the extouch triangle are: a'= sqrt(a^2 - 4*A^2/b*c) b'= sqrt(b^2 - 4*A^2/a*c) c'= sqrt(c^2 - 4*A^2/a*b) where A is the triangle area of the original triangle. The extouch triangle has area: 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. It is interesting to note that the sides of the extouch triangles are irrational numbers (in the general case) but the areas are integers. 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. ------------------------------------------------------------- A'  |   A  | a | b  | c  |     t1    |     t2    |  t3 ------------------------------------------------------------- 30  |  150 | 15| 20 | 25 | 3*sqrt(5) | 4*sqrt(10)|5*sqrt(13) 48  |  300 | 25| 25 | 40 | sqrt(265) | sqrt(265) | 32 72  |  300 | 25| 25 | 30 | sqrt(145) | sqrt(145) | 18 84  | 1050 | 35| 75 |100 | 7*sqrt(13)|3*sqrt(385)|8*sqrt(130) 120 |  600 | 30| 40 | 50 | 6*sqrt(5) |8*sqrt(10) |10*sqrt(13) 192 | 1200 | 50| 50 | 80 |2*sqrt(265)|2*sqrt(265)| 64 252 | 2100 | 35|120 |125 |    7      |72*sqrt(2) |5*sqrt(457) 270 | 1350 | 45| 60 | 75 |9*sqrt(5)  |12*sqrt(10)|15*sqrt(13) 288 | 1200 | 50| 50 | 60 |2*sqrt(145)|2*sqrt(145)| 36 336 | 4200 | 70|150 |200 |14*sqrt(13)|6*sqrt(485)|16*sqrt(130) 432 | 2700 | 75| 75 |120 |3*sqrt(265)|3*sqrt(265)| 96 480 | 2400 | 60| 80 |100 |12*sqrt(5) |16*sqrt(10)|20*sqrt(13) 648 | 2700 | 75| 75 | 90 |3*sqrt(145)|3*sqrt(145)| 54 .................................................. 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). 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. ------------------------------- A'  |  A   |  h1 |  h2 |  h3 ------------------------------- 30  |  150 |  20 |  15 |  12 48  |  300 |  24 |  24 |  15 72  |  300 |  24 |  24 |  20 84  | 1050 |  60 |  28 |  21 120 |  600 |  40 |  30 |  24 192 | 1200 |  48 |  48 |  30 252 | 2100 | 120 |  35 | 168/5 270 | 1350 |  60 |  45 |  36 288 | 1200 |  48 |  48 |  40 336 | 4200 | 120 |  56 |  42 432 | 2700 |  72 |  72 |  45 480 | 2400 |  80 |  60 |  48 648 | 2700 |  72 |  72 |  60 ............................... LINKS T. Dosa, Some Triangle Centers Associated with the Excircles, Forum Geometricorum, Volume 7 (2007) 151-158. C. Kimberling, Triangle Centers and Central Triangles, Congr. Numer. 129, 1-295, 1998. Eric Weisstein's World of Mathematics, Extouch Triangles EXAMPLE 30 is in the sequence. We use two ways: First way: the formula A'= A*(a+b-c)*(a-b+c)*(-a+b+c)/4abc 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. Second way: by calculation of the sides t1, t2, t3 and by using Heron's formula. 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: a' = sqrt(a^2 - 4*A^2/b*c) = sqrt(15^2-4*150^2/(20*25)) = 3*sqrt(5); b' = sqrt(b^2 - 4*A^2/a*c) = sqrt(20^2-4*150^2/(15*25)) = 4*sqrt(10); c' = sqrt(c^2 - 4*A^2/a*b) = sqrt(25^2 - 4*150^2/(15*20)) = 5*sqrt(13). Now, we use Heron's formula with (t1,t2,t3). We find A'=sqrt(s1*(s1-t1)*(s1-t2)*(s1-t3))with: s1 =(t1+t2+t3)/2 = (3*sqrt(5)+ 4*sqrt(10) + 5*sqrt(13))/2; We find A'= 30. MATHEMATICA 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] CROSSREFS Cf. A188158, A210643. Sequence in context: A004222 A207143 A004223 * A272949 A271998 A228097 Adjacent sequences:  A230555 A230556 A230557 * A230559 A230560 A230561 KEYWORD nonn AUTHOR Michel Lagneau, Oct 23 2013 STATUS approved

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Last modified January 18 10:53 EST 2019. Contains 319271 sequences. (Running on oeis4.)