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.

(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 $$\cos^{-1}\left(\frac{r}{\sqrt{r^2+s^2}}\right)+\cos^{-1}\left(\frac{r+s}{\sqrt{2}\sqrt{r^2+s^2}}\right) = \frac{\pi}{4}$$.

News

Summer Session 2014

Link to Summer Session class, which starts May 19 and ends June 27
Welcome to the Fall semester of 2019!
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 2020).

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.

Talk on GeoGebra at Missouri MAA Section Meeting

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.

MAA PREP Professional Development Workshop

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.