TITLE: What novel results are there in Euclidean geometry in the last 50-100 years?
QUESTION [10 upvotes]: Most summaries of the history of geometry conclude with an overview of developments in non-Euclidean geometry over the 18th-20th century. Some mention reformulations of Euclidean geometry based on different postulates with similar implications, more sophisticated treatments might mention algebraic approaches. But I'm wondering what new theorems (ideally proven via Euclidean techniques) there may be in the last century or so, or if there are current questions practitioners may be pursuing answers to.

REPLY [3 votes]: Here are a couple examples involving the properties of polygons:

We can divide a square region into smaller square regions that are all of different size.  The first published division dates to the 1930s.

It is readily proven, in both Euclidean and non-Euclidean geometries, that the area of a convex polygon inscribed in a circle depends only on the lengths of its sides and not on how the sides are ordered.  Quantifying the area is another matter; for five or more sides the high-degree polynomial equations that govern the area began to be uncovered only by Robbins in the 1990s.  Wolfram Mathworld, in a brief article on the pentagonal case, cites this work [1,2].


References
1.
Robbins, D. P. "Areas of Polygons Inscribed in a Circle." Discr. Comput. Geom. 12, 223-236, 1994.
2.
Robbins, D. P. "Areas of Polygons Inscribed in a Circle." Amer. Math. Monthly 102, 523-530, 1995.