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

TL;DR

This paper provides a synthetic proof of the tangency property of the parbelos by introducing a converse to Lambert's Theorem on the parabola, along with new properties of the parbelos.

Contribution

It offers a novel synthetic proof of Sondow's tangency property and introduces a converse to Lambert's Theorem on the parabola.

Findings

Proof of the tangency property of the parbelos.

Introduction of a converse to Lambert's Theorem.

New properties of the parbelos.

Abstract

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.