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

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

Offset: 1

Author

Michel Lagneau, Oct 23 2013

Keywords

• nonn

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

...............................

Examples

			30 is in the sequence. We use two ways:
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.
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.
		

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

Crossrefs

Cf. A188158, A210643.

Programs

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]