A while back I developed a mild obsession with pentagons (mathematical ones, not symbolistic!) It started when I discovered some beautiful (simple and to me, unknown) theorems of quadrangles, such as Varignon’s theorem. I already came across Miquel’s pentagon theorem, and wondered what other gems I could find.

Here is what I found: Pentagons (2.4 MB PDF).

My search was on the surface a bit disappointing: pentagons as such are not widely studied. I guess it is because some more general theorems (that apply to general polygons) contain the theory, and the specifics as applied to pentagons are not so interesting.

Nevertheless, I did discover a few theorems, and the journey took me into some very interesting corners of geometry; a very rewarding experience. I started to collect these into a document, which is shared below. It’s not comprehensive or complete; there are a few gaps.

(At some stage I may return to look at this again. In particular, there are many theorems of the type “if there is n, there is n+1”, which seems to me to hint at a very general theorem which can be used to prove a bunch of specifics.)

Also, when I started, I did not realise how many of the theorems will generalise to general polygons, so the collection looks a bit silly in retrospect (kind of like listing all the properties of the number 8 that equally apply to even numbers).

Even so, what’s done is done, and can perhaps satisfy someone else’s curiosity.

Here is the list of contents:

- Notation
- Standard labeling
- Cycle notation (
*I introduce a convention that to me makes it easier to keep track of symbols in my head.)* - Area

- Five points in a plane
- General Pentagons
- Monge, Gauss, Ptolemy (
*Theorems that have analogues for quadrangles. For example, one theorem relates the areas of six triangles spanned by certain vertices of the pentagon.*) - Cyclic Ratio Products (alla Ceva en Melenaus) (
*Theorems involving products of cevians and other ratios.*) - Miquel (
*Thereoms analoguous to Miquel’s theorem for triangles.*) - Conics (
*Circumscribed and inscribed conics are uniquely determined by a pentagon.*) - Complete Pentagons (
*Includes Miquel’s pentagram theorem.*) - The Centroid Theorem (
*A general theorem about centroid applied to pentagons.*)

- Monge, Gauss, Ptolemy (
- Special Pentagons (
*Including how to construct these pentagons.*)- Cyclic Pentagons (
*Pentagons with vertices on a circle.*) - Tangent Pentagons (
*Pentagons with all sides tangent to a circle.*) - Orthocentric Pentagons (
*Pentagons whose altitudes intersect in a single point.*) - Mediocentric Pentagons (
*Pentagons whose medians intersect in a single point.*) - Paradiagonal Pentagons (
*Pentagons whose diagonals are parallel to oposite sides. Also called golden pentagons, affine regular pentagons, and equal area pentagons.*) - Equilateral Pentagons (
*Pentagons with all sides of equal length.*) - Equiangular Pentagons (
*Pentagons with all interior angles equal.*) - Brocard Pentagons (
*Pentagons that have a Brocard point.*) - Classification By Subangles (
*What the relationships between subangles imply for the pentagon.*)

- Cyclic Pentagons (