Constructing a curve of constant width from mutually intersecting lines On this website http://kmoddl.library.cornell.edu/math/2/ it is given that any curve constructed in the following manner will be a curve of constant width. 
"Draw as many straight lines as you please all mutually intersecting. Each arc is drawn with the compass point at the intersection of the two lines that bound the arc. Start with any arc, then proceed around the curve, connecting each arc to the preceding one. If you do it carefully, the curve will close and will have a constant width."

Rather infuriatingly, the source then goes on to say that this fact is easy to prove on your own, yet does not offer a proof or sketch of a proof for this remarkable construction. How might a proof of this fact proceed, and what theorems would it invoke in the process?
 A: 
A side-arc $\stackrel{\frown}{AB}$ is determined by two lines that also determine an "opposite" side-arc, $\stackrel{\frown}{A^\prime B^\prime}$. For a point $P$ on $\stackrel{\frown}{AB}$, there is a corresponding point $P^\prime$ on $\stackrel{\frown}{A^\prime B^\prime}$ such that $\overline{PP^\prime}$ passes through the intersection of those lines. Since $\overline{PP^\prime}$ is perpendicular to the tangents to the side-arcs at $P$ and $P^\prime$, its length —the sum of the radii of those arcs— is the figure's width, and thus that width is constant across the entirety of the side-arcs. Likewise for the "next" side-arcs $\stackrel{\frown}{BC}$ and $\stackrel{\frown}{B^\prime C^\prime}$. Since vertices $B$ and $B^\prime$ are shared by across arcs, the "next" side-arcs inherit the sum-of-radii, and so forth to subsequent side-arcs. Consequently, the figure (presuming the arcs have been chosen "carefully", as the description says, to form a closed loop) has constant width. $\square$
Note: In this answer to OP's previous question, I demonstrate that the curvature of neighboring side-arcs of the Reuleaux triangle do not interfere with the curve being constant width. Here, the argument is straightforward: the line containing a pair of opposite vertices is perpendicular to the tangents to the side-arcs adjacent to those vertices; thus, the vertices (locally) maximize the width of the figure.
