# Euclid

## t-Formulae and parameterisation of the circle

t-Formulae are used in integration to tackle rational expressions of tigonometric functions.  After a spell in the cold, when they were not included in some A level specifications, they are now back in sixth form lessons. It all starts with the subsitution, from which the following functions can be derived, ,  ,  . These derivations …

## Area of a circle – proof by exhuastion

Once has been defined as the ratio circumference to diameter, the area of a circle must be . A proof relies on an infinite, limiting process which paves the way to some calculus-like ideas. A circle is cut up into 6 sectors which are then rearranged into a near rectangle, What doesn’t look too close …

## Parallelograms generalise Pythagoras to any triangle

There’s always space on the inter-web for another proof of Pythagoras’s Theorem.  Here’s one that uses the following equal areas property of parallelograms. This kind of area chopping and shape translation is a feature of Euclidean geometry and our senses support it’s veracity at the order of size of the classroom. The squares on the …

## Before the beginning – Euclid’s Common Notions

This preamble from Euclid’s Elements is where mathematics education still goes wrong, even after 2000 years or so. Sound notions of what equality is are required as a bedrock of algebra.  Dynamic approaches of teaching equation solving, in which terms and numbers move and change sign, seem to work at first but lack the simplicity …

## First Definition of Triangles

First definition of triangles from Euclid’s Elements Book I (Fundamentals of Plane Geometry Involving Straight Lines), Definition 20.

## The Angle in a Semi-Circle, Euclid Book III, Proposition 31

Ancient theorem of Euclid (Book III, Proposition 31) still a fundamental part of school maths.  The angle in a semi-circle is a right angle.