Notice

In the pages that are reached through the links below, geometric diagrams are no longer drawn.

Why?

Those geometric drawing relied on David Joyce's Geometry Applet which in turn relied on Java applets. After Oracle's acquisition of Sun in 2010, Oracle dropped applets from Java, and thus the Geometry Applet stopped working.

That's unfortunate. One of these days when I find some free time, I may add static (non-interactive) goemetric drawings to supplement the geometry pages. For now, you may read those pages and attempt to construct the missing diagrams from their verbal descriptions.

However...

In February 2020 I began converting the interactive illustrations to GeoGebra which is based on javascript and is supported quite well. Some of the puzzles have already been converted. As time permits, I will convert more, in the order listed on this page. When all is done, this Notice will go away.

Geometry Problems and Puzzles

This page contains a motley collection of problems and puzzles of elementary geometry that at one point or another have intrigued and amused me. I hope that you will enjoy them as much as I have.

Wherever possible, I have used David Joyce's wonderful Geometry Applet to produce interactive illustrations and diagrams. You will need a java-enabled browser to see the drawings.

List of problems and puzzles

The asymmetric propeller: A theorem popularized by Martin Gardner in an article in The College Journal of Mathematics.
Common tangents of two circles: A relationship between the common tangents of two circles.
Equal chords in two circles: An application of the “power of a point” theorem.
The maximal angle problem: On how to get the widest angle of view.
The rings of Saturn: Where to stand to get the best view of the rings?
An inequality concerning equilateral triangles: …and a nice solution by Dan Hoey.
Another problem concerning equilateral triangles: Intriguing puzzle proposed by Steve Gray.
An equilateral triangle inscribed in a rectangle: Maybe easy to solve… I don't know.
A triangle's incenter via algebra: … and extension to tetrahedra.

Angle trisection

There are a surprisingly large number of clever geometric constructions that achieve approximate trisections of arbitrary angles within the classical straightedge-and-compass restrictions. I have gathered a few samples here. Each sample is accompanied with an illustration, step-by-step description of the constructions, and error analysis. I have ordered them in the table below in accordance with the method's precision; the more precise methods being closer to the top.

I measure a method's precision by its worst error in trisecting angles in the range 0 to 90 degrees. Thus, the error in the method proposed by Free Jamison is no more than 0.00067 degrees, which is orders of magnitude smaller than many competitors.

For those constructions that work for angles up to 180 degrees, I have also listed the maximum trisection error over the range 0 to 180 degrees, otherwise I have entered “NA”.

**Click on the links to see the details**
Author	Max err. in degrees in the 0–90 range	Max err. in degrees in the 0–180 range
Chris Alberts	1.33×10^-16 ^(*)	NA
Mark Stark	0.00013 ^(**)	0.0592 ^(**)
Free Jamison	0.00067	0.00537
Avni Pllana	0.0434	0.361
C. R. Lindberg	0.0434	0.361
R. L. Durham	0.152	1.252
Wayne Baker	0.154	1.360
A. G. O.	0.171	1.367
William R. Raiford	0.361	3.435
cdsmith	1.23	1.23
Anonymous	1.352	NA

(*) The extraordinary precision of Chris Alberts' construction is obtained through twice application of an iterative improvement. Should the improvment be applied only once, its precision would be quite clsoe to that of Mark Stark's.

(**) Mark Stark's construction involves a free choice of a point. The construction quality depends on the choice of that point. The values given in the table above correspond to a “natural” choice; see the link for details.

Author: Rouben Rostamian