Let $L_0,L_1,L_2,L_3,L_4$ be five lines in general position on the Euclidean plane---think of the subscripts mod $5$ and draw $L_i$ as the consecutive lines of a (not necessarily regular!) pentagram. Let $C_i$ be the circle inscribed about the triangle formed by $L_i$, $L_{i-2}$, and $L_{i+2}$. Then $C_{i-1}$ and $C_{i+1}$ meet at the intersection of $L_{i-1}$ and $L_{i+1}$, and again at some other point $P_i$ (which we take to be the same point if the two circles are tangent there). Show that these five points $P_i$ are concyclic.