A230402 Integer areas of orthic triangles of integer-sided triangles.

1512, 2688, 6048, 10752, 13608, 24192, 37800, 43008, 54432, 67200, 74088, 96768, 122472, 131712, 151200, 172032, 182952, 217728, 255528, 268800, 296352, 325248, 340200, 387072, 436968, 454272, 489888, 526848, 545832, 604800, 666792, 688128, 731808, 776832

Offset

1,1

Comments

Given a triangle ABC, the triangle H1H2H3 whose vertices are endpoints of the altitudes from each of the vertices of ABC is called the orthic triangle, or sometimes the altitude triangle. The three lines AH1, BH2, and CH3 are concurrent at the orthocenter H of the triangle ABC.

The area of the orthic triangle is given by S'= abc|cos A cos B cos C|/2R where a, b, c are the integer sides of ABC and R is the circumradius of ABC.

The lengths of the legs of the orthic triangle are given by

a' = a|cos A| = a|b^2+c^2-a^2|/2bc;

b' = b|cos B| = b|c^2+a^2-b^2|/2ac;

c' = c|cos C| = c|a^2+b^2-c^2|/2ab.

Property of this sequence:

It seems that all triangles are isosceles of the form (a, a, b) with a < b, and each associated orthic triangle is also of the form (h1, h1, h2) but with h1> h2(see the table below). It seems also that if the area of the orthic triangle is an integer, then the sides are integers.

The following table gives the first values (S’, S, a, b, c, h1, h2, h3) where S' is the area of the orthic triangles, S is the area of the triangles ABC, a, b, c the integer sides of the triangles ABC and h1, h2, h3 are the integer sides of the orthic triangles.

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

* S’ * S * a * b * c * h1 * h2 * h3 *

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

* 1512 * 7500 * 125 * 125 * 150 * 75 * 75 * 42 *

* 2688 * 7500 * 125 * 125 * 200 * 100 * 100 * 56 *

* 6048 * 30000 * 250 * 250 * 300 * 150 * 150 * 84 *

* 10752 * 30000 * 250 * 250 * 400 * 200 * 200 * 112 *

* 13608 * 67500 * 375 * 375 * 450 * 225 * 225 * 126 *

* 24192 * 67500 * 375 * 375 * 600 * 300 * 300 * 168 *

* 24192 * 120000 * 500 * 500 * 600 * 300 * 300 * 168 *

* 37800 * 187500 * 625 * 625 * 750 * 375 * 375 * 210 *

* 43008 * 120000 * 500 * 500 * 800 * 400 * 400 * 224 *

* 54432 * 270000 * 750 * 750 * 900 * 450 * 450 * 252 *

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

References

H. S. Coxeter and M. Greitzer, The Orthic Triangle, §1.6 in Geometry Revisited, Washington DC, Math. Assoc. Amer., pp. 9 and 16-18, 1967.

R. Honsberger, The Orthic Triangle, §2.3 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Washington DC, Math. Assoc. Amer., pp. 21-25, 1995.

Links

Table of n, a(n) for n=1..34.

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

Eric W. Weisstein, MathWorld: Orthic Triangle

Formula

Empirical g.f.: -168*x*(9*x^10 -2*x^9 +22*x^8 +6*x^7 +11*x^6 +52*x^5 +11*x^4 +6*x^3 +22*x^2 -2*x +9) / ((x -1)^3*(x +1)*(x^2 -x +1)^2*(x^2 +x +1)^2). - Colin Barker, Oct 22 2013

Example

1512 is in the sequence because the orthic triangle (h1, h2, h3) associated with the triangle (a, b, c) = (125, 125, 150) is the triangle (75, 75, 42) where:
a' = a|cos A| = a|b^2+c^2-a^2|/2bc = 75;
b' = b|cos B| = b|c^2+a^2-b^2|/2ac = 75;
c' = c|cos C| = c|a^2+b^2-c^2|/2ab = 42.
and the area is given by Heron's formula: sqrt((96*(96-75)*(96-75)*(96-42)) = 1512.
It is possible to use also the formula S'= abc|cos A cos B cos C|/2R = h1*h2*h3/2R with R = a*b*c/4S where S is the area of the triangle (a, b, c) and S' the area of the orthic triangle. Hence S' = h1*h2*h3*2*S/a*b*c = 75*75*42*2*7500/(125*125*150) = 1512.

Mathematica

nn=200; 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))/(4*(a^2*b^2*c^2))]; If[0<area2 && aa>0 && IntegerQ[Sqrt[area2]*aa], AppendTo[lst, Sqrt[area2]*aa]]], {a, nn}, {b, a}, {c, b}]; Union[lst]

Crossrefs

Sequence in context: A200915 A298223 A060676 * A248718 A255334 A255778

Adjacent sequences: A230399 A230400 A230401 * A230403 A230404 A230405

Keyword

nonn

Author

Michel Lagneau, Oct 18 2013

Status

approved