Solution of Sondow's problem: a synthetic proof of the tangency property of the parbelos

In a recent paper titled "The parbelos, a parabolic analog of the arbelos", Sondow asks for a synthetic proof to the tangency property of the parbelos. In this paper, we resolve this question by introducing a converse to Lambert's Theorem on the parabola and in the process prove some new properties of the parbelos.