The Various Radii Circle Packing Problem in a Triangle

Malfatti’s problem is the problem of fitting three circles into a triangle such that they are tangent to each other and each circle is also tangent to a pair of the triangle’s sides. This problem has been extended to include Tn = 1 + 2 + … + n circles inside the triangle with special tangency properties among the circles and triangle sides; this problem is referred to as the extended Malfatti problem or the Tri(Tn) problem. In the extended Malfatti problem, the number of circles in the triangle is a triangle number because the tangency properties between the internal circles and the three sides of the triangle have a special type of structure; that is, the corner circle is tangent to two sides of the triangle and two other circles, the boundary circles are tangent to one side of the triangle and four other circles, and the inner circles are always tangent to six other circles. The circles we find in the extended Malfatti problem have the following property: the smallest and largest radii of the circles differ to a great extent. In the study presented herein, we propose algorithms to solve the problem that the tangency properties between the circles and the sides of the triangle are not fixed, so that the number of circles in the triangle is not necessarily a triangle number. The purpose of this change is to attempt to establish the radii of the circles in the triangle within a small range.