A couple of years ago I wrote blog posts about two beautiful theorems from geometry: the so-called Japanese theorem and Carnot’s theorem. Today I finished a draft of a web article that looks at both of these theorems in more detail. It contains all that you could want—connections between these theorems, generalizations of them, and consequence of…
Tag: geometry
Albrecht Dürer’s ruler and compass constructions
Albrecht Dürer (1471–1528) is a famous Renaissance artist. Mathematicians probably know him best for his work Melencolia I which contains a magic square, a mysterious polyhedron, a compass, etc. Today I was reading his book Underweysung der Messung mit dem Zirckel und Richtscheyt (The Painter’s Manual: A manual of measurement of lines, areas, and solids…
Lincoln and squaring the circle
I’d heard a long time ago that Abraham Lincoln was a largely self-taught man and that he read Euclid’s Elements on his own. Right now I’m reading Doris Kearns Goodwin’s Team of Rivals: The Political Genius of Abraham Lincoln, and from it I learned that not only did he read Euclid, he spent some time…
Three geometric theorems
Just for fun I thought I’d share a few interesting geometric theorems that I came across recently. Morley’s miracle In 1899 Frank Morley, a professor at Haverford, discovered the following remarkable theorem. The three points of intersection of the adjacent trisectors of the angles of any triangle form an equilateral triangle. I’ve made a Geogebra…
Circle squaring limerick
I found this nice limerick on Charles Petzold’s blog: Said the man about town, ‘I have a flair For squaring the circle, I swear.’ But he found that the strain Was too great for his brain, So he’s gone back to circling the square. Petzold has a scan of the title page of E. H….
An Euler line geogebra applet
Lately I’ve been thinking a lot about Euler and his many contributions to mathematics. So, just for fun I decided to make a Geogeba applet showing the “Euler Line.” In 1763 Euler proved that three different centers of a triangle—the centroid, the orthocenter, and the circumcenter—are collinear. This line is called the Euler line. The centroid…
The geometry of meandering rivers
I regularly watch the Science Friday video podcast. This week they had an interesting piece on potamology (OK, I just learned that word and wanted to use it in my post: potamology is the scientific study of rivers). The podcast showcased the work Christian Braudrick and Bill Dietrich of University of California, Berkeley, who achieved…
Kindergarten mathematics
This is a call for help. My son’s kindergarten teacher has invited parents to come in and talk about their careers. I’d like to go in and talk about math. I’d like to have some interactive hands-on mathematics activities for the kids to do. I also want them to be activities outside the typical kindergarten…
Carnot’s Theorem
Here’s a neat theorem from geometry. Begin with any triangle. Let R be the radius of its circumscribed circle and r be the radius of its inscribed circle. Let a, b, and c be the signed distances from the center of the circumscribed circle to the three sides. The sign of a, b, and c…
The Japanese Theorem
[Update: I’ve written quite a bit more about this theorem since 2009. See this page for more details.] I’ve been playing with GeoGebra for the last few days. As an exercise I decided to create applets to demonstrate the extremely beautiful Japanese Theorem. The first appearance of the Japanese theorem was as a Sangaku problem….
David Richeson: Division by Zero 