18.27 A point inside a convex quadrilateral is called a "good point" if the lines connecting it to the four vertices divide the quadrilateral into four triangles of equal area.
Proposition A: There exists a "good point" inside the quadrilateral.
Proposition B: One diagonal of the convex quadrilateral bisects the other diagonal. Then the correct statement is
(A) A is a sufficient but not necessary condition for B.
(B) B is a sufficient but not necessary condition for A.
(C) A is a necessary and sufficient condition for B.
(D) B is neither a sufficient nor a necessary condition for A.
(Anhui Province Wuhu City High School Mathematics Competition, 1990)
[Solution] If $O$ is a "good point" inside the convex quadrilateral $ABCD$, and let $E, F$ be the midpoints of diagonals $AC, BD$ respectively.
From $S_{\triangle OAB}=S_{\triangle OBC}$, we know that point $O$ should lie on $BE$.
Similarly, we find that point $O$ should also lie on $CF, DE, AF$, in other words, $O$ is the intersection of $BE, CF, DE, AF$.
Now we prove that lines $AF$ and $CF$, or $BE$ and $DE$ coincide.
If $AF$ and $CF$ do not coincide, then $O$ is their only common point, so $O$ coincides with $F$.
Since lines $BE, DE$ pass through $O$ which is also $F$, and $F$ lies on $BD$, then $BE, DE$ and $BD$ coincide. That is, $B, E, D$ are collinear, so $BD$ passes through the midpoint of diagonal $AC$.
Conversely, if the diagonal $BD$ of quadrilateral $ABCD$ is bisected by $AC$, it is easy to prove that
$$
S_{\triangle ABE}=S_{\triangle AED}, S_{\triangle BCE}=S_{\triangle DCE},
$$

Then the midpoint $O$ of $AC$ is a "good point".
$$
\begin{array}{cc}
\because & S_{\triangle ABO}=S_{\triangle BCO}, \\
\text { and } & S_{\triangle ABO}=S_{\triangle ABE}+S_{\triangle BEO}, \\
\text { also } & S_{\triangle BEO}=S_{\triangle EDO}, \quad \therefore \quad S_{\triangle ABO}=S_{\triangle ADO}, \\
\because & O \text { is the midpoint of } AC, \therefore \quad S_{\triangle ADO}=S_{\triangle DOC}, \\
\therefore & S_{\triangle BOC}=S_{\triangle AOB}=S_{\triangle AOD}=S_{\triangle COD} .
\end{array}
$$

Thus, proposition A is a necessary and sufficient condition for proposition B.
Therefore, the answer is (C).