# Generalization of a property of scalene trapezoids

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?

• What's a simple polygon?
– user239203
Aug 3, 2020 at 12:58
• With simple I mean that the boundaries of the polygon do not cross itself Aug 3, 2020 at 12:59
• Then I don't understand the first paragraph: all triangles are simple (also all convex polygons are simple).
– user239203
Aug 3, 2020 at 13:00
• Yes, but this is not necessarily true for polygons with more sides Aug 3, 2020 at 13:02
• The main point of my question addresses the loss of "being" a special case of n-1 sided polygons if 1 point is removed from a n sided polygon Aug 3, 2020 at 13:04