A230361 Integer areas of the tangential triangles corresponding to the integer-sided triangles with integer areas.

23660, 26250, 53235, 94640, 105000, 147875, 212940, 222530, 236250, 378560, 390390, 420000, 479115, 591500, 656250, 788970, 851760, 945000, 1286250, 1330875, 1561560, 1680000, 1916460, 2126250, 2608515, 2625000, 3176250, 3407040, 3513510

Offset

1,1

Comments

The tangential triangle is the triangle (A', B', C') of side lengths (a', b', c') formed by the lines tangent to the circumcircle of a given triangle (A, B, C) of side lengths (a, b, c) at its vertices.

The area of the tangential triangle is given by Area = (1/2)*Area(A,B,C)*|sec A * sec B * sec C|

The side lengths of the tangential triangle are:

a' = 2*a^3*b*c/|a^4 - (b^2 - c^2)^2|

b' = 2*a*b^3*c/|b^4 - (c^2 - a^2)^2|

c' = 2*a*b*c^3/|c^4 - (a^2 - b^2)^2|

In the general case, these sides lengths are rational numbers (see the examples in the table below). But it is possible to find integer sides, for example the triangle (210, 210, 252) generates a tangential triangle (625, 625, 350).

It is possible to find the same tangential triangle with two distinct triangles of side lengths (a1,b1,c1) and (a2,b2,c2); for example, the triangles (105, 105, 175) and (140, 140, 224) generate the same tangential triangle (625/2, 625/2, 175).

The following table gives the first values (S', S, a, b, c, a', b', c') where S' is the area of the tangential triangle (A', B', C'), S is the area of the initial triangle (A, B, C), a, b, c the integer sides of the triangle (A, B, C) and a', b', c' are the integer sides of the tangential triangle (A', B', C').

**************************************************************

S' * S * a * b * c * a' * b' * c'

**************************************************************

23660 * 10584 * 84 * 273 * 315 * 182 * 2197/6 * 1625/6

26250 * 5292 * 105 * 105 * 126 * 625/2 * 625/2 * 175

26250 * 9408 * 140 * 140 * 224 * 625/2 * 625/2 * 175

26250 * 24192 * 168 * 360 * 480 * 175 * 625/2 * 625/2

53235 * 8064 * 104 * 160 * 168 * 2197/4 * 1625/4 * 273

94640 * 42336 * 168 * 546 * 630 * 364 * 2197/3 * 1625/3

105000 * 21168 * 210 * 210 * 252 * 625 * 625 * 350

105000 * 37632 * 280 * 280 * 448 * 625 * 625 * 350

147875 * 40320 * 200 * 416 * 504 * 8125/12 * 10985/12 * 455

212940 * 32256 * 208 * 320 * 336 * 2197/2 * 1625/2 * 546

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

References

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929.

Kimberling, C. Triangle Centers and Central Triangles. Congr. Numer. 129, 1-295, 1998.

Links

Table of n, a(n) for n=1..29.

C. Kimberling, Central Points and Central Lines in the Plane of a Triangle, Math. Mag. 67, 163-187, 1994.

Eric W. Weisstein, MathWorld: Tangential Triangle

Example

26250 is in the sequence because the triangle of integer sides (a, b, c)= (105, 105, 126) generates the tangential triangle (a', b', c') = (625/2, 625/2, 175) where:
a' = 2*a^3*b*c/|a^4 - (b^2 - c^2)^2| = 625/2
b' = 2*a*b^3*c/|b^4 - (c^2 - a^2)^2| = 625/2
c' = 2*a*b*c^3/|c^4 - (a^2 - b^2)^2| = 175
The area of this triangle is given by two possible ways:
First way:
Heron's formula gives Area = sqrt(s'*(s'-a')*(s'-b')*(s'-c')) = sqrt(400*(400-625/2)*(400-625/2)*(400-175)) = 26250 with the semiperimeter s' = (625/2 + 625/2 + 175)/2 = 400.
Second way:
Area of the triangle (a, b, c) = sqrt(sqrt(s*(s-a)*(s-b)*(s-c))=sqrt(168*(168-105)*(168-105)*(168-126))=5292 with the semiperimeter s = (105 + 1052 + 126)/2 = 168.
Then, we use the formula Area = (1/2)* Area(A,B,C) * |sec A * sec B * sec C| = 2646*5/3*5/3*25/7 = 26250 where:
sec A = 1/cos A = 2*b*c/(b^2+c^2-a^2)= 5/3;
sec B = 1/cos B = 2*a*c/(c^2+a^2-b^2)= 5/3;
sec C = 1/cos C = 2*a*b/(a^2+b^2-c^2)= 25/7.

Mathematica

nn=1500; lst={}; Do[s=(a+b+c)/2; If[IntegerQ[s], area2=s (s-a) (s-b) (s-c); aa=Abs[((b^2+c^2-a^2)*(c^2+a^2-b^2)*(a^2+b^2-c^2))]; If[0 < area2 && aa>0&& IntegerQ[Sqrt[area2]* (4*a^2*b^2*c^2)/aa], AppendTo[lst, Sqrt[area2]* (4*a^2*b^2*c^2)/aa]]], {a, nn}, {b, a}, {c, b}]; Union[lst]

Crossrefs

Cf. A188158.

Sequence in context: A237508 A177827 A304652 * A069334 A118061 A251855

Adjacent sequences: A230358 A230359 A230360 * A230362 A230363 A230364

Keyword

nonn

Author

Michel Lagneau, Oct 16 2013

Status

approved