G87 (47-2, 塞黑) Let $P$ be a regular 2006-gon. If an end of a diagonal of $P$ divides the boundary of $P$ into two parts, each containing an odd number of sides of $P$, then the diagonal is called a "good edge". It is stipulated that each side of $P$ is a "good edge".

Given 2003 non-intersecting diagonals inside $P$ that divide $P$ into several triangles, how many isosceles triangles with two "good edges" can there be in such a division?



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G87 (47-2, 塞黑) Let $P$ be a regular 2006-gon. If an end of a diagonal of $P$ divides the boundary of $P$ into two parts, each containing an odd number of sides of $P$, then the diagonal is called a "good edge". It is stipulated that each side of $P$ is a "good edge".

Given 2003 non-intersecting diagonals inside $P$ that divide $P$ into several triangles, how many isosceles triangles with two "good edges" can there be in such a division?
If an isosceles triangle has two good sides, it is briefly referred to as an "iso-good" triangle.
Let $\triangle A B C$ be an "iso-good" triangle, and $A B, B C$ be the good sides. Then, there are an odd number of edges between $A$ and $B$, and the same applies between $B$ and $C$. We say these edges belong to the "iso-good" triangle $A B C$.

In each of these groups, at least one edge does not belong to any other "iso-good" triangle. This is because two vertices between $A$ and $B$ in an "iso-good" triangle have two equal-length sides, thus there are an even number of edges belonging to it. By removing all edges that belong to any other "iso-good" triangle, there must be one edge left that does not belong to any other "iso-good" triangle. Designate these two edges (one in each group) to belong to $\triangle A B C$. For each "iso-good" triangle, designate a pair of edges, ensuring no two triangles share the designated edges. It follows that under this partition, there can be at most 1003 "iso-good" triangles. It is easy to draw a partition that achieves this value.