16. (This question is worth 12 points)\\
This question has three optional parts I, II, and III. Please select any two to answer and write your solutions in the corresponding answer areas on the answer sheet. If you answer all three, only the first two will be graded.\\
I (This question is worth 6 points) Elective 4-1: Geometric Proof\\
As shown in Figure 5, in circle O, two chords AB and CD intersect at point E, with midpoints M and N respectively. The line MO intersects line CD at point F. Prove:\\
(I) $\angle \mathrm { MEN } + \angle \mathrm { NOM } = 180 ^ { \circ }$;\\
(II) $\mathrm { FE } \cdot \mathrm { FN } = \mathrm { FM } \cdot \mathrm { FO }$

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{a6eae9e4-32c9-40f4-807d-a64508802a24-4_426_521_591_370}
\captionsetup{labelformat=empty}
\caption{Figure 5}
\end{center}
\end{figure}

II. (This question is worth 6 points) Elective 4-4: Coordinate Systems and Parametric Equations\\
Given the line $l: \left\{ \begin{array} { l } x = 5 + \frac { \sqrt { 3 } } { 2 } t \\ y = \sqrt { 3 } + \frac { 1 } { 2 } t \end{array} \right.$ (where t is the parameter). With the origin as the pole and the positive x-axis as the polar axis, the polar equation of curve C is $\rho = 2 \cos \theta$\\
(i) Convert the polar equation of curve C to rectangular coordinates;\\
(II) Let the rectangular coordinates of point M be $( 5 , \sqrt { 3 } )$. The line $l$ intersects curve C at points $A$ and $B$. Find the value of $| M A | \cdot | M B |$\\
III. (This question is worth 6 points) Elective 4-5: Inequalities\\
Let $\mathrm { a } > 0$, $\mathrm { b } > 0$, and $\mathrm { a } + \mathrm { b } = \frac { 1 } { a } + \frac { 1 } { b }$. Prove\\
(i) $\mathrm { a } + \mathrm { b } \geqslant 2$;\\
(ii) $\mathrm { a } ^ { 2 } + \mathrm { a } < 2$ and $\mathrm { b } ^ { 2 } + \mathrm { b } < 2$ cannot both be true.