# geometry

## 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 …

## 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 …

## 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.

Enlargement.

## IGCSE Transformations – Rotation

Rotations in space make good Saturday morning maths at the King’s School.

## Great Circles and Spherical Triangles

In the geometry of mariners, where straight lines are Great Circles, the angle sum of a triangle is more than 180 degrees.

This method for finding the centre, and by extension the equation, of a circle given three non-colinear points, brings the ancient textbook master Euclid onto Descartes’ coordinate plane and right  into the 21 century classroom.

jpedmaths: Angles in the same segment are equal. A reminder for my IGCSE students.  Name the theorem correctly; even though you might understand the maths, ‘donkey’s ears’ is not an exam sufficient tag.