5.109. a) Let $B_{1}, C_{1}$, and $D_{1}$ be the projections of point $P$ onto the lines $A B, A C$, and $A D$. The points $B_{1}, C_{1}$, and $D_{1}$ lie on a circle with diameter $A P$. The lines $B_{1} C_{1}, C_{1} D_{1}$, and $D_{1} B_{1}$ are the Simson lines of point $P$ with respect to the triangles $A B C, A C D$, and $A D B$, respectively. Therefore, the projections of point $P$ onto the Simson lines of these triangles lie on a single line - the Simson line of the triangle $B_{1} C_{1} D_{1}$. Similarly, it can be shown that any triple of the considered points lies on a single line.

b) Let $P$ be a point on the circumcircle of the $n$-gon $A_{1} \ldots A_{n}$; $B_{2}, B_{3}, \ldots, B_{n}$ be the projections of point $P$ onto the lines $A_{1} A_{2}, \ldots, A_{1} A_{n}$. The points $B_{2}, \ldots, B_{n}$ lie on a circle with diameter $A_{1} P$. We will prove by induction that the Simson line of point $P$ with respect to the $n$-gon $A_{1} \ldots A_{n}$ coincides with the Simson line of point $P$ with respect to the $(n-1)$-gon $B_{2} \ldots B_{n}$ (for $n=4$ this was proven in part a). By the induction hypothesis, the Simson line of the $(n-1)$-gon $A_{1} A_{3} \ldots A_{n}$ coincides with the Simson line of the $(n-2)$-gon $B_{3} \ldots B_{n}$. Therefore, the projections of point $P$ onto the Simson lines of the $(n-1)$-gons, whose vertices are obtained by sequentially excluding points $A_{2}, \ldots, A_{n}$ from the set $A_{1}, \ldots, A_{n}$, lie on the Simson line of the $(n-1)$-gon $B_{2} \ldots B_{n}$. And the projection of point $P$ onto the Simson line of the $(n-1)$-gon $A_{2} \ldots A_{n}$ lies on the same line because our reasoning shows that any $n-1$ of the considered $n$ projection points lie on a single line.