Question 109
Let $n \gt 2$ and $A_1, A_2, ..., A_n$ be $n$ points on the plane that no three of them are collinear.
a. Let $M_1, M_2, ..., M_n$ be $n$ points on the segments $A_1A_2, A_2A_3, ..., A_nA_1$ respectively. Show that if $B_1, B_2, ..., B_n$ are $n$ points in the triangles $M_nA_1M_1, M_1A_2M_2, ..., M_{n-1}A_nM_n$ respectively, then the following holds
$$
| B_1B_2 | + | B_2B_3 | + \dots + | B_nB_1 | \le | A_1A_2 | + | A_2A_3 | + \dots + | A_nA_1 |
$$
where $|XY|$ means the length of the segment $XY$.
b. We define $H_{XYZ}$ as the half-plane with the outer bisector of angle $\overline{XYZ}$ as its boundary which does not contain the inner bisector of $\overline{XYZ}$.
Show that if $C_1, C_2, ..., C_n$ are $n$ points in the half-planes $H_{A_nA_1A_2}, H_{A_1A_2A_3}, ..., H_{A_{n-1}A_nA_1}$ respectively, then
$$
| A_1A_2 | + | A_2A_3 | + | A_nA_1 | \le | C_1C_2 | + | C_2C_3 | + \dots + | C_nC_1 |
$$
a. Let $M_1, M_2, ..., M_n$ be $n$ points on the segments $A_1A_2, A_2A_3, ..., A_nA_1$ respectively. Show that if $B_1, B_2, ..., B_n$ are $n$ points in the triangles $M_nA_1M_1, M_1A_2M_2, ..., M_{n-1}A_nM_n$ respectively, then the following holds
$$
| B_1B_2 | + | B_2B_3 | + \dots + | B_nB_1 | \le | A_1A_2 | + | A_2A_3 | + \dots + | A_nA_1 |
$$
where $|XY|$ means the length of the segment $XY$.
b. We define $H_{XYZ}$ as the half-plane with the outer bisector of angle $\overline{XYZ}$ as its boundary which does not contain the inner bisector of $\overline{XYZ}$.
Show that if $C_1, C_2, ..., C_n$ are $n$ points in the half-planes $H_{A_nA_1A_2}, H_{A_1A_2A_3}, ..., H_{A_{n-1}A_nA_1}$ respectively, then
$$
| A_1A_2 | + | A_2A_3 | + | A_nA_1 | \le | C_1C_2 | + | C_2C_3 | + \dots + | C_nC_1 |
$$