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,

(1)

where is the circumradius, is the inradius, and is the semiperimeter. For side lengths , this condition is also equivalent to

(2)

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

			(3)
			(4)
			(5)

where are integers. Its semiperimeter and area are

			(6)
			(7)

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

The Pythagorean quadruple yields the same family by taking the pairwise sums of squares of its first three components.