login
A276072
Radius of the outer Soddy circle (or the longest side) of primitive Heronian triangles whose outer Soddy center lies on its longest side.
1
6, 21, 52, 95, 105, 186, 259, 273, 301, 392, 456, 549, 603, 657, 790, 910, 1001, 1023, 1067, 1133, 1221, 1308, 1596, 1651, 1677, 1729, 1807, 1911, 2041, 2114, 2282, 2535, 2562, 2715, 2985, 3088, 3165, 3216, 3472, 3689, 3723, 3791, 3856, 3893, 4029, 4199, 4403, 4446, 4641, 4662
OFFSET
1,1
COMMENTS
It has been shown that if the outer Soddy circle center of a triangle lies on one of its sides then this side is the longest and is equal in length to both the outer Soddy radius and the excircle radius associated with the vertex opposite the longest side. Furthermore, the sum of the tangents of the triangle's half angles opposite the two smaller sides equals 1.
LINKS
F. M. Jackson, S. Takhaev, Heronian Triangles of Class J: Congruent Incircles Cevian Perspective, IJCDM, 1/3 (2016), 1-8.
EXAMPLE
The triangle ABC with sides {a, b, c} = {21, 20, 13} has an outer Soddy radius and an exradius (opp. vertex A) equal to 21. Its area is 126. The outer Soddy center lies on BC (= 21) at the tangential point with the A excircle at length 15 from B.
MATHEMATICA
a=2n(m^2+3n^2); b=(m+n)(m^2-2m*n+5n^2); c=-(m-n)(m^2+2m*n+5^2); lst={}; Do[If[GCD[m, n]==1, AppendTo[lst, a/GCD[a, b, c]]], {n, 1, 100}, {m, 0, n-1}]; Take[Sort@lst, 50] (* Corrected by Frank M Jackson, May 12 2017 *)
CROSSREFS
Sequence in context: A097124 A309568 A244906 * A135454 A069778 A015644
KEYWORD
nonn
AUTHOR
Frank M Jackson, Stalislav Takhaev, Sep 10 2016
EXTENSIONS
Trailing terms corrected by Frank M Jackson, May 12 2017
B-file corrected by Giovanni Resta, May 13 2017
STATUS
approved