Some theorems on kissing circles and spheres (Q2912904)

Let \(O_1, O_2\) and \(O_3\) be circles externally tangent to each other, and suppose that the circle \(S\) contains \(O_1,O_2\) and \(O_3\) in its interior and is tangent to all of \(O_1, O_2\) and \(O_3\). Let \(I_1, I_2\) and \(I_3\) be the three additional circles tangent to \(S, O_2\) and \(O_3\), to \(S, O_1\) and \(O_3\) and to \(S, O_1\) and \(O_2\), respectively. Denote by \(s_i\) and \(t_i\) the two common external tangents of \(O_i\) and \(I_i\), \(i=1,2,3\). Then there exists a circle tangent to all of \(s_1, t_1, s_2, t_2, s_3\) and \(t_3\). Moreover, let \(\triangle A_1 A_2 A_3\) denote the triangle whose edges are given by the outer common external tangents of \(O_1,O_2\) and \(O_3\) and, similarly, let \(\triangle B_1 B_2 B_3\) be given by the outer common external tangents of \(I_1,I_2\) and \(I_3\). Finally, let \(l_1,l_2\) and \(l_3\) be the straight lines joining opposite vertices of \(\triangle A_1 A_2 A_3\) and \(\triangle B_1 B_2 B_3\). Then \(l_1,l_2\) and \(l_3\) pass through a common point.NEWLINENEWLINEThe elementary proof of the above is based on inversions of circles and spheres. An analogous claim is shown for three-dimensional tangent spheres.