Let A,B,C be vertices of a triangle T= T(A,B,C) in the Euclidean plane. The inscribed circle is the largest circle inside T, which touches it at exactly three points A',B',C'. Continuing iteratively we get a nested sequence of triangles which converge to a unique point z. J.L.Synge (1897-1995) posed the problem of understanding the dependence of z = z(A,B,C) on the initial triangle. We shall discuss this and some generalizations (e.g., to higher dimensions)