# Geometry

## A Triangle Problem: With Trig, and Without

(A new question of the week) When you are given a problem about a triangle, there can be many ways to approach it: pure geometry, trigonometry, and analytic geometry come to mind. When the context doesn’t dictate a method (as turns out to be true here), you just have to try what feels right to …

## Average Distance Between Two Sets of Points

(A new question of the week) Here we have a different kind of question than usual: A conjecture about distances between points, with a request for confirmation. Normally we like to just give hints to help a student figure something out; this was a request for a theorem that ought to exist, and trying to …

## Application of Vectors: Airplane in the Wind

A recent question about the resultant velocity of an airplane illustrates different ways to make a diagram showing the bearings of air velocity and wind velocity, and to work out angles without getting too dizzy.

## Distance from a Point to a Line in Space

(A new question of the week) Among many interesting recent questions we have one about vectors and equations in three dimensions. We’ll see four different ways to find the distance from a point to a line, proving two formulas and catching some of the errors one can make along the way. We’ll also see a …

## Graph Coloring: Working Through a Proof

(A new question of the week) The Math Doctors have different levels of knowledge in various fields; I myself tend to focus on topics through calculus, which I know best, and leave the higher-level questions to others who are more recently familiar with them. But sometimes, both here and in my tutoring at a community …

## The Golden Ratio and Fibonacci

We’re looking at the Fibonacci sequence, and have seen connections to a number called phi (φ or $$\phi$$), commonly called the Golden Ratio. I want to look at some geometrical connections and other interesting facts about this number before we get back to the Fibonacci numbers themselves and some inductive proofs involving them.

## Angles in a Star

(A new question of the week) I like problems that can be solved in multiple ways, which can train us in seeing the world from different perspectives. Late in November we dealt with a pair of such questions involving angles in star-like figures.

## Disappearing Area?

We’ve been looking at dissection puzzles, where we cut an object into pieces, and rearrange them. Here we’ll examine a mystery posed by two different puzzles, each of which seems to change the area by rearranging the pieces. The answer combines the marvelous Fibonacci numbers and [spoiler alert!] how easily we misjudge areas.

## Cutting and Rearranging a Rectangle

Last week we looked at a puzzle about cutting a square cake into equal pieces. Here we will be trying to cut a rectangle into two pieces and rearranging them to make a different rectangle. Three of the questions we’ll look at came within two weeks in 2001, but we’ll take them in a logical …

## A Geometrical Limit

(A new question of the week) We usually see limits applied to functions in a calculus class. An interesting question from late October deals with a limit in a geometrical construction based on a function. We’ll be seeing how to discover a proof, then several alternative proofs, and finally what the answer means.