## Fun Problem of the Week/Month/Semester

Verify the claims made in the tweet below by first determining the lengths of the non-hypotenuse sides of the dissection triangles.

Interesting dissection of the 345 triangle showing radius of inscribed circle is 1 and arctan(1)+arctan(1/2)+arctan(1/3)=pi/2 pic.twitter.com/w5F1BONn3L

— Michael Blaylock (@MickBlaylock) February 7, 2018

(Source: Twitter; accessed February 8, 2018)

Extension: show for any (real) \(r > s \geq 0\) that \(\tan^{-1}\left(\frac{s}{r}\right)+\tan^{-1}\left(\frac{r-s}{r+s}\right) = \frac{\pi}{4}\).

(Use the parameterization \(a=2rs, b=r^2-s^2, c=r^2+s^2\) (where \(r > s > 0\)) for an \(a\)-\(b\)-\(c\) right triangle. The identity can also be proved directly using analytic geometry, or by using an appropriate triangle inscribed within a rectangle of dimensions \((r+s) \times r\).)

This identity says that if we take two right triangles of (respective) bases \(r\) and \(r+s\) and (respective) heights \(s\) and \(r-s\) (with respective hypotenuses \(\sqrt{r^2+s^2}\) and \(\sqrt{2}\sqrt{r^2+s^2}\)), then the acute angles of these triangles add to \(\frac{\pi}{4}\), and that two more identities hold, namely \(\sin^{-1}\left(\frac{s}{\sqrt{r^2+s^2}}\right)+\sin^{-1}\left(\frac{r-s}{\sqrt{2}\sqrt{r^2+s^2}}\right) = \frac{\pi}{4}\) and \(\cos^{-1}\left(\frac{r}{\sqrt{r^2+s^2}}\right)+\cos^{-1}\left(\frac{r+s}{\sqrt{2}\sqrt{r^2+s^2}}\right) = \frac{\pi}{4}\).

(Use the parameterization \(a=2rs, b=r^2-s^2, c=r^2+s^2\) (where \(r > s > 0\)) for an \(a\)-\(b\)-\(c\) right triangle. The identity can also be proved directly using analytic geometry, or by using an appropriate triangle inscribed within a rectangle of dimensions \((r+s) \times r\).)

This identity says that if we take two right triangles of (respective) bases \(r\) and \(r+s\) and (respective) heights \(s\) and \(r-s\) (with respective hypotenuses \(\sqrt{r^2+s^2}\) and \(\sqrt{2}\sqrt{r^2+s^2}\)), then the acute angles of these triangles add to \(\frac{\pi}{4}\), and that two more identities hold, namely \(\sin^{-1}\left(\frac{s}{\sqrt{r^2+s^2}}\right)+\sin^{-1}\left(\frac{r-s}{\sqrt{2}\sqrt{r^2+s^2}}\right) = \frac{\pi}{4}\) and \(\cos^{-1}\left(\frac{r}{\sqrt{r^2+s^2}}\right)+\cos^{-1}\left(\frac{r+s}{\sqrt{2}\sqrt{r^2+s^2}}\right) = \frac{\pi}{4}\).

## News

**Link to Summer Session class, which starts May 19 and ends June 27**

**Welcome to the Spring semester of 2018!**

Use the links in the panels on the left to navigate this site.

Use the links in the panels on the left to navigate this site.

### SLU Inquiry Seminar

**On-line brochure for Math 1250 Mathematical Thinking in the Real World (taught Spring 2018).**

### Updated List of Mathematics Blogs and Links on "Mathematics links" Page

**Featured links: What is it like to understand advanced mathematics? and Habits of highly mathematical people.**

**I gave a talk titled A Brief Tour of GeoGebra at the 2012 Missouri MAA Section Meeting held at the University of Missouri St. Louis April 12-14.**

**Michael May, S.J. and I ran an online workshop during Summer 2011: Web-Enhanced Instruction with Geogebra, July 11 - July 15, 2011.**

### A Resource for Strengthening Math Skills

**mathmistakes.info is an external web site I am developing that explains common mistakes made by students in classes from algebra to multivariable calculus, and provides online flashcards and other resources for strengthening math skills used in those classes.**