# Geometry

## 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.

## Cutting a Square Cake Equally

For the next couple weeks, we’ll look at a few “dissection problems”, which involve cutting a shape in various ways. Here, we look at what turns out to be a simple problem, though it seems at first that it would be complicated: dividing a cake into pieces with the same amount of cake and the …

## Finding an Angle With and Without Trig

(A new question of the week) Usually when we have a figure labeled with some lengths and angles, we can expect to find unknown angles using trigonometry. When we are expected to do this using geometry alone, we can expect that there is something special about the figure that makes it possible. But how to …

## Broken Sticks, Triangles, and Probability I

This week we look at questions about how likely it is that you can make a triangle out of three random pieces of a stick. As always in probability, the first issue comes in deciding how the process is to be done (that is, what does it mean to break a stick randomly?); we’ll also …

## Infinitely Truncating a Triangle

(A new question of the week) Here is an intriguing question we got at the end of September from an adult whose name I’ll shorten to Arun. Dear Sir Let Po be an equilateral triangle of area 10. Each side of Po is trisected, and the corners are snipped off, creating a new polygon (in …

## A Proof Problem: Chords and Tangents

One thing we enjoy doing is guiding a student through the process of problem-solving. Here is a problem from August that illustrates how to think through a complicated geometrical proof. In particular, this uses some circle theorems involving chords, secants, and tangents, together with a bit of algebra. The problem Here is the problem (I’ve …