Proof

Trig Identities: Where’d That Come From?

(A new question of the week) Proving a trigonometric identity can be a challenge; sometimes even when we read someone else’s proof, we can fail to see how they came up with a seemingly magical step. We’ll look at two such identities here, and consider how to bridge a gap when we are stuck.

What is Mathematical Induction?

Last week’s exploration of a problem involving the Fibonacci sequence, has led me to delve into that and related concepts. In order to say much about the Fibonacci numbers, we have to first explore the concept of proof by mathematical induction. We’ll introduce it here, and then dig deeper next time.

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.

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.