Let $ABCD$ be any quadrilateral, and let $r, s, t, u$ be its four angle bisectors. If the $r,s,t,u$ are not concurrent (as in the image below on the left), then they intersect to form the vertices of another quadrilateral $PQRS$ (as in the image below on the right).
I don't know if there is a standard name for this construction; let's call $PQRS$ the "angle bisector quadrilateral" or abq of $ABCD$.
Here are some fairly simple properties:
- The angle bisectors are concurrent if and only if $ABCD$ is a quadrilateral in which $AB + CD = BC + AD$.
- As a special case, if $ABCD$ is a kite (including the special case of a rhombus) then the angle bisectors are concurrent.
- If the angle bisectors are not concurrent, then the abq is a cyclic quadrilateral -- i.e. one in which the opposite angles are supplementary.
- If $ABCD$ is a parallelogram (but not a rhombus) then its abq is a rectangle.
- If $ABCD$ is a rectangle (but not a square) then its abq is a square.
In particular if we start with a parallelogram and iterate the construction we have the following sequence: $$\textrm{parallelogram} \to \textrm{rectangle} \to \textrm{square} \to \textrm{point}$$
This suggests the following questions:
- Is it true that for any quadrilateral, iterating the construction of abqs leads eventually to a single point after finitely many iterations?
- If so, is there an upper bound on the number of iterations?
- If no, is it possible to have a sequence of iterations that is periodic (up to scale) in the sense that the $n$th iterate is similar to the origin figure?
I'd be interested in thoughts on these questions, even if they are inconclusive.