Astute geometry students know that when a problem involves a 3 and a 4, the answer will involve a 5 and the solution will make use of a right triangle somehow. The triple of numbers $(3,4,5)$ is called a Pythagorean… Continue Reading

## Three Circles in a Triangle: A Solution

In my last post, I posed the problem of showing that if triangles $ABC$ and $PQR$ (as shown in the diagram below) are similar, then $\Delta ABC$ is equilateral. We will solve this problem in two steps: Show that the… Continue Reading

## Three Circles in a Triangle: A Problem

This week, I was playing around with some geometry and came up with a fun problem. Suppose there is a triangle, $\Delta ABC$. Inside the triangle are three circles, each of which is tangent to the other circles and to… Continue Reading

## Pythagorean Days

Those of you who have been obsessing about today’s solar eclipse may have missed another equally amazing event just last Tuesday: the date, 8/15/17, made a Pythagorean triple. A Pythagorean triple is a group of three positive integers that satisfy… Continue Reading

## A Near Miss

Recently, I was working with my son on some of his Beast Academy homework on tangrams. Tangrams are a set of seven tiles (5 triangles of 3 different sizes along with a square and a parallelogram) that can be arranged to… Continue Reading

## Puffy Triangles

A few months ago, I was working on a lesson on sectors of circles. A sector of a circle is a piece of it cut off by slicing along two radiuses from the center. There is a formula for the… Continue Reading

## Self-Diagram

A few weeks ago, I stumbled across this gem of a geometry problem from an old MATHCOUNTS exam: The intersection of a circular region of radius $3$ inches and a circular region of radius $2$ inches has area $\pi$ square… Continue Reading

## Happy Approximation to Pi Day!

March 14 is one of my least favorite days of the year. While most math teachers revel in Pi Day, I spend my 3/14s frustrated that we waste an opportunity to pay homage to one of the most important numbers in… Continue Reading

## Bagel Math

The Three Utility Problem is a classical mathematical puzzle: Three customers, Alice, Bob, and Charlie, all want to connect their houses to three different utilities, Electricity, Gas, and Water. Can each customer run a line to each utility company in… Continue Reading

## Hyperbolic Space

In Spring of 2016, I taught a new elective at IMSA called Modern Geometries. The course started with a deep dive into Euclid’s Elements followed by a careful analysis of the consequences when some of Euclid’s (often unstated) premises were challenged.… Continue Reading