Limits of iterated geometric constructions

In triangle $A B C_{1}$, the bisector $C_{1} C_{2}$ is drawn; in triangle $A C_{1} C_{2}$, the bisector $C_{2} C_{3}$ is drawn; in triangle $A C_{2} C_{3}$, the bisector $C_{3} C_{4}$, and so on. Prove that the sequence of measures of the angles $C_{n+1} C_{n} A, n \in \mathbb {N}$, has a limit, and find it, if the angle $B A C$ equals $\alpha$.

The circle inscribed in triangle $A_{1} B_{1} C_{1}$ touches its sides $B_{1} C_{1}, C_{1} A_{1}$, and $A_{1} B_{1}$ at the points $A_{2}, B_{2}, C_{2}$, respectively; the circle inscribed in triangle $A_{2} B_{2} C_{2}$ touches its sides $B_{2} C_{2}, C_{2} A_{2}, A_{2} B_{2}$ at the points $A_{3}, B_{3}, C_{3}$, respectively, and so on. Find the limit of the sequence of measures of the angles $B_{n} A_{n} C_{n}$ as $n \rightarrow \infty$.

In trapezoid $A B C D$ $(A D \parallel B C)$, the points $B_{1}$ and $C_{1}$ are the midpoints of the diagonals $A C$ and $B D$; in trapezoid $A B_{1} C_{1} D$, the points $B_{2}$ and $C_{2}$ are the midpoints of the diagonals $A C_{1}$ and $B_{1} D$, and so on. Find $\lim_{n \rightarrow \infty }\left|B_{n} C_{n}\right|$, if $\left|A D\right| = a$, in the cases: а) $\left|A D\right| > \left|B C\right|$; б) $\left|A D\right| < \left|B C\right|$.

In triangle $A B C$, medians $A D, B E$, and $C F$ are drawn. On the side $A C$, a point $M_{0}$ is taken, $M_{0} \neq A, M_{0} \neq C$, and from it a point $M_{1}$ on $A C$ is found as follows: segments $\left[M_{0} N_{0}\right] \parallel [C F] \left(N_{0} \in A B\right)$ are drawn, then $\left[N_{0} P_{0}\right] \parallel [A D] \quad \left(P_{0} \in B C\right)$, and finally $\left[P_{0} M_{1}\right] \parallel [B E]$. Starting from the point $M_{1}$, the point $M_{2}$ is found analogously, and so on.

Prove that the sequence of points $\left\{ M_{n}\right\}$ converges, and find its limit.

Let $P_0 P_1 P_2$ be a triangle right-angled at $P_2$, $P_2 P_3$ the perpendicular from $P_2$ to $P_0 P_1$, similarly $P_3 P_4 \perp P_1 P_2$, $P_4 P_5 \perp P_2 P_3$, etc. Find the point $\lim_{n \to \infty } P_n$.