A228435 Integer areas of the mixtilinear triangles of integer-sided triangles.

6, 14, 24, 54, 56, 84, 96, 112, 124, 126, 130, 150, 154, 156, 210, 216, 224, 266, 294, 336, 342, 344, 350, 366, 384, 436, 448, 456, 486, 496, 504, 520, 531, 546, 570, 600, 616, 624, 630, 660, 686, 726, 756, 814, 840, 864, 896, 924, 1008, 1014, 1064, 1116, 1134

Offset: 1

Author

Michel Lagneau, Nov 10 2013

Keywords

• nonn

Comments

The mixtilinear triangle is the triangle connecting the centers of the mixtilinear incircles.

A circle that in internally tangent to two sides of a triangle and to the circumcircle is called a mixtilinear incircle. There are three mixtilinear incircles, one corresponding to each angle of the triangle.

The area of the mixtilinear triangle is given by

A' = A*(a^3 - a^2*b - a*b^2 + b^3 - a^2*c + 6*a*b*c - b^2*c - a*c^2 - b*c^2 + c^3)/(a + b + c)^3

where A is the area of the reference triangle of sides (a, b, c).

See the link for the side lengths of the mixtilinear triangles.

Properties of this sequence:

The primitive mixtilinear triangles are 6, 14, 84, 112, 124, 130, ...

The nonprimitive triangles of areas 4*a(n), 9*a(n), ..., p^2*a(n), ... are in the sequence.

It appears that if the triangles are isosceles, one of the sides of the mixtilinear triangles is an integer, for example, for the triangles of areas a(n) = 112, 342, 448, 1008, 1368, 1792, 2800, 3078, 4032, 5472, ... corresponding to a triangle with an integer side of values respectively 8, 24, 16, 24, 48, 32, 40, 72, 48, 96, ...

The following table gives the first values (A', A, a, b, c, a', b', c') where A' is the area of the mixtilinear triangles, A is the area of the initial triangles, a, b, c are the integer sides of the initial triangles, and a', b', c' are the sides of the mixtilinear triangles.

-------------------------------------------------------------------------------

A' | A| a| b| c| a' | b' | c'

-------------------------------------------------------------------------------

6 | 54| 9|12| 15| 9*sqrt(5)/4 | 4*sqrt(13)/3 | 5*sqrt(37)/12

14 | 126|15|28| 41|135*sqrt(1717)/196 |28*sqrt(757)/27 | 41*sqrt(16045)/5292

24 | 216|18|24| 30| 9*sqrt(5)/2 | 8*sqrt(13)/3 | 5*sqrt(37)/6

56 | 504|30|56| 82|135*sqrt(1717)/98 |56*sqrt(757)/27 | 41*sqrt(16045)/2646

84 | 756|39|42| 45|104*sqrt(445)/147 | 7*sqrt(37)/3 | 45*sqrt(197)/49

96 | 864|36|48| 60| 9*sqrt(5) |16*sqrt(13)/3 | 5*sqrt(37)/3

112| 972|45|45| 72| 20*sqrt(2) |20*sqrt(2) | 8

124|1116|39|62| 85|351*sqrt(8605)/961 |31*sqrt(853)/27 |680*sqrt(79957)/25947

126|1134|45|84|123|405*sqrt(1717)/196 |28*sqrt(757)/9 | 41*sqrt(16045)/1764

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

Examples

			6 is in the sequence. We use two ways:
First way: from the initial triangle (9, 12, 15) the formula given in the comments gives directly the area of the mixtilinear triangle: A' = 54*(9^3 - 9^2*12 - 9*12^2 + 12^3 - 9^2*15 + 6*9*12*15 - 12^2*15 - 9*15^2 - 12*15^2 + 15^3)/(9 + 12 + 15)^3 = 6, where the area of the initial triangle A = 54 is obtained by Heron's formula A = sqrt(s*(s-a)*(s-b)*(s-c)) = sqrt(18*(18-9)*(18-12)*(18-15)) = sqrt(2916) = 54, where s=18 is the semiperimeter.
Second way: by calculation of the sides a', b', c' and by use of Heron's formula. With the formulas given in the link, we find
a’ = 9*sqrt(5)/4;
b’ = 4*sqrt(13)/3;
c’ = 5*sqrt(37)/12.
Now, we use Heron's formula with (a',b',c'). We find A' = sqrt(s1*(s1-a')*(s1-b')*(s1-c')) with:
s1 = (a' + b' + c')/2 = (9*sqrt(5)/4 + 4*sqrt(13)/3 + 5*sqrt(37)/12)/2.
We find A' = 6.
		

Links

• Eric Weisstein's World of Mathematics, Mixtilinear triangles

Crossrefs

Cf. A188158.

Programs

Mathematica

nn = 500; lst = {}; Do[s = (a + b + c)/2; t = a^3 - a^2*b - a*b^2 + b^3 - a^2*c + 6*a*b*c - b^2*c - a*c^2 - b*c^2 + c^3; If[IntegerQ[s], area2 = s (s - a) (s - b) (s - c); If[0 < area2 && IntegerQ[Sqrt[area2] * t/(a + b + c)^3], AppendTo[lst, Sqrt[area2] * t/(a + b + c)^3]]], {a, nn}, {b, a}, {c, b}]; Union[lst]