The Gergonne and Soddy lines (Q2341521)

Given three pairwise externally tangent circles \(\mathcal{A}\), \(\mathcal{B}\) and \(\mathcal{C}\) centred at \(A\), \(B\) and \(C\), the Soddy circles \(\mathcal{S}\) and \(\mathcal{S}'\) are the two other circles that are tangent to these three at points \(T\), \(U\) and \(V\). The Soddy line is the line passing through the centers of the Soddy circles. The Gergonne line is the line from which the triangles \(\triangle ABC\) and \(\triangle TUV\) are in perspective. A pencil of circles (the so-called Gergonne pencil) is constructed and some properties are investigated in the previous setting.