Parallelograms

If, instead of considering the intersections of adjacent angle trisectors of a triangle, we now consider the four intersection points of the adjacent angle trisectors of a parallelogram, what do we get? The figures below show some possible situations, with the initial parallelogram represented in gray and the polygon obtained by consecutively joining the intersection points of the angle trisectors (when these points are not collinear) marked in green.

The analysis of the figures suggests that, independently of the initial parallelogram, the obtained polygon is always another parallelogram (except when the points are collinear). Check this with the help of the following interactive applet:

(Click on the points \(A\), \(B\) and \(D\) to change the shape of the parallelogram \([ABCD]\))

In particular, when the initial parallelogram is a rectangle, the obtained polygon appears to be a rhombus, and conversely, when the initial parallelogram is a rhombus, the obtained polygon appears to be a rectangle. In fact, all these conjectures are true. Similar to what happens with the triangles, the parallelogram obtained by the intersection of adjacent angle trisectors relating to two consecutive vertices of an initial parallelogram is here called Morley's parallelogram.

Of course, starting from a square, we always get a smaller square (a square is both a rectangle, because all its angles are right angles, and a rhombus, because all sides are equal). More generally, if we start from any regular polygon, we always obtain other regular polygons with the same number of sides, only smaller ( I wonder why?). You can see this property with the following applet:

(Move the blue point to change the number of vertices and the red points to change the size and position of the polygon)

Note that other choices for the intersection points of angle trisectors also produce interesting results. For example, even in the case of the parallelogram, instead of choosing the intersection points of adjacent angle trisectors corresponding to two consecutive vertices, we could have chosen the intersection points of the other two angle trisectors concerning consecutive vertices. In this case we can also conclude that these points are either colinear or are the vertices of another parallelogram, which is also a rhombus when the initial parallelogram is a rectangle and vice versa. In the applet below the initial parallelogram is represented in gray and the parallelograms obtained by the intersection of its angle trisectors are represented in green and purple.

(Click on the points \(A\), \(B\) and \(D\) to change the shape of the parallelogram \([ABCD]\))