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GeezerVison is showing mathworld.wolfram.com — Soddyian Triangle -- from Wolfram MathWorld

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Soddyian Triangle

A Soddyian triangle is a triangle whose outer Soddy circle degenerates into a common tangent line of the three mutually tangent circles centered at its vertices (Jackson 2013, pp. 1-2). Equivalently,

4R+r=2s,
(1)

where R is the circumradius, r is the inradius, and s is the semiperimeter. For side lengths a<=b<=c, this condition is also equivalent to

1/(sqrt(s-c))=1/(sqrt(s-a))+1/(sqrt(s-b)).
(2)

Every integer-sided Soddyian triangle is a Heronian triangle (Jackson 2013, p. 3). A family is given by

a=n^2[(m+n)^2+m^2]
(3)
b=m^2[(m+n)^2+n^2]
(4)
c=(m+n)^2(m^2+n^2),
(5)

where m>=n>0 are integers. Its semiperimeter and area are

s=(m^2+mn+n^2)^2
(6)
Delta=m^2n^2(m+n)^2(m^2+mn+n^2).
(7)

For example, (m,n)=(1,1) gives the isosceles triangle with side lengths (5, 5, 8) and area 12, while (m,n)=(2,1) gives side lengths (13, 40, 45) and area 252 (Jackson 2013, pp. 3-4).

The Pythagorean quadruple (mn,n(m+n),m(m+n),m^2+mn+n^2) yields the same family by taking the pairwise sums of squares of its first three components.

See also

Heronian Triangle, Outer Soddy Circle, Pythagorean Quadruple, Soddy Circles

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References

Jackson, F. M. "Soddyian Triangles." Forum Geometricorum 13, 1-6, 2013. https://www.researchgate.net/publication/265504822_Soddyian_triangles.

Cite this as:

Weisstein, Eric W. "Soddyian Triangle." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SoddyianTriangle.html

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