Consider a scalene trapezoid, which is a special case of simple (no crossing boundaries) quadrilaterals (since it has two parallel sides). When one removes a vertex from this trapezoid and reconnects the remaining vertices using 2 old lines and the trapezoid's diagonal, one obtains a triangle. However, this triangle is not necessarily a special case of simple 3-sided polygons (i.e. it is not necessarily isosceles or equilateral).
So, a scalene trapezoid has the property that a simply polygon constructed from 3 of its vertices might be the most general simple 3-sided polygon (triangle), while it itself is a special case of simple 4-sided polygons (since it has two parallel sides).
My question is: Are there any other polygons with more than 4 vertices that have the same property? E.g. is there a special case of a simple octagon (parallel sides, pair of identical angles,...) that with one vertex removed and reconnected may result in a general simple heptagon?
Does this property have a name?