Welcome! The purpose of this blog is to record some of the particularly beautiful mathematical ideas I have seen or invented, and share them with you.

The process of doing mathematics is much like a quest to uncover mathematical truths. Sometimes, such a truth may be valid but uninteresting, just another pebble or grain of sand along the beach. But other times, you will uncover a gemstone – a particularly aesthetic, beautiful, or useful truth hiding in the vast sandpiles of information.

This blog is devoted to the gemstones of my mathematical investigations. I hope you enjoy it!

**Ratings: ** The posts are rated in difficulty according to five categories: Amber, Pearl, Opal, Sapphire, and Diamond. (Notice that they are ordered by the hardness of the gemstones.)

**Amber:**This category contains posts that anyone with very basic elementary or middle school mathematics background can appreciate.**Pearl:**For Pearl posts, some high school courses or early college courses may be helpful in understanding the content.**Opal:**These posts are aimed at undergraduates with some basic first-course background knowledge in algebra, analysis, discrete math, or topology.**Sapphire:**These gemstones would be appreciated by mathematics graduate students or professors, or those with at least an undergraduate degree in mathematics.**Diamond:**The hardest type of gemstone. These posts are highly specialized, containing content that only mathematicians who have studied the topic in depth will have the background to understand.

# About the Author

Dr. Maria Monks Gillespie is a mathematician at Colorado State University in Fort Collins, Colorado. She got her start in mathematics through success in math competitions in high school, and went on to get a bachelors’ in mathematics at MIT and a Ph.D. at the University of California, Berkeley. Dr. Gillespie’s specific research interests lie in algebraic combinatorics, but she loves diving into any area of math and discovering its secrets.

She also is one of the head instructors – along with her parents, brothers, and husband – at Prove it! Math Academy, a two-week residential summer program in Colorado designed for talented high school students looking to make the transition from computational problem solving to research and proof-based mathematics.

# Acknowledgments

Thank you to Daniel Matriccino for helping me get this off the ground by hosting the first version of this website on his servers, which also host the new outdoor activity organization website terrahubs.com.

Thank you also to Ken G. Monks for showing me the magic html header lines needed to get MathJax rendering integrated into my blog, and for setting up the server that hosts proveitmath.org, on which this blog is also currently hosted.

Thanks to my husband Bryan Gillespie for his good eye and helpful feedback on the banner design, and my brother Keenan Monks for suggesting some of the gemstones to represent various levels of difficulty.

Finally, thanks to readers like you for making my writing worthwhile!

Two cevians ideas I have found useful

Draw a diameter of the circumcircle perpendicular to the base. Since the base is a chord the diameter will bisect it. The apex bisector is perpendicular to the line from the apex to the intersection of the constructed diameter with the top of the circle. Extend the bisector to the intersection of the lower end of the diameter at the bottom of the circle. The proof that it is the bisector is trivial because the intercepted arc are equal.

The symmedian is Apollonius’ sub-contrary axis so it passes through the center of the anti-parallel to the base. The anti-parallel is parallel to the tangent to the circumcircle at the apex, or alternately, perpendicular to the line from the center of the circumcircle to the apex.

I have a small html5 GeoGebra model that runs in the browser if you would like it and tell me how to get it to you.

Hi, is it OK for me to use a photo i found on your site called Railway Track? I would like to use it to explain 1-point perspective in a drawing book I am updating, You Too Can Draw. Kindest regards, Nicola Sedgwick

Hi Nicola,

I’m not sure which photo you’re referring to; it doesn’t appear in my blog media collection. I may have put such a photo up in an old version but decided not to use it because I wasn’t sure if there were copyrights on the photo. If so, the picture is not mine, so I cannot say.

Best,

Maria

Hello Maria. This is completely unrelated to the topic addressed here but how can I create an account in this website?

Thank you.