18. 14 (IMO 43 Pre) For any set $S$ of 5 points in the plane, such that no three points in $S$ are collinear. Let $M(S)$ and $m(S)$ be the maximum and minimum areas of triangles formed by any 3 points in $S$, respectively. Find the minimum value of $M(S): m(S)$.
18. 14 When these five points are the vertices of a regular pentagon, \( M(S): m(S) \) equals \( \tau = \frac{1 + \sqrt{5}}{2} \), which is the desired minimum value.

Let the five points in \( S \) be \( A, B, C, D, E \), and the area of \( \triangle ABC \) be \( M(S) \). It can be proven by construction that there exists a triangle whose area is less than or equal to \( M(S) : \tau \).

Construct \( \triangle A' B' C' \) such that each side is parallel to the corresponding side of \( \triangle ABC \), and \( A, B, C \) are the midpoints of \( B' C', C' A', A' B' \), respectively. Then, points \( D \) and \( E \) must be inside or on the sides of \( \triangle A' B' C' \). Otherwise, it would contradict the assumption that \( \triangle ABC \) has the maximum area. Assume without loss of generality that \( D \) and \( E \) are inside the quadrilateral \( B' B' C' \).

Since the ratio of areas \( M(S): m(S) \) remains unchanged when the figure \( S \) undergoes an affine transformation [an affine transformation is a one-to-one correspondence of points in a plane established by a parallel projection. Parallelism and collinearity are affine invariants; the cross ratio of three collinear points, the ratio of parallel segments, and the area ratio of corresponding figures are affine invariants; a parallelogram is an affine invariant figure.], we can first apply an affine transformation to \( A, B, C \) to meet certain requirements; then, discuss the possible positions of \( D \) and \( E \), and complete the proof using the method of affine transformation.