## 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 sides of a right-angle triangle set up a system of parallel lines which can then be used to demonstrate the Theorem using the above equal areas property. The thread does not stop here though.  Taking the parallel line structure which makes this work we get a generalisation of Pythagoras to non-right-angled triangles with the area of the parallelogram on the longest side being the sum of the areas of those constructed on the other two sides. ## 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 of mathematical logic and create all sorts of problems when randomly, but apparently sensibly, applied.

## 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.

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.

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.

Some things don’t change.  This Theorem of Euclid’s, now IGCSE geometry, has probably been taught at The King’s School Canterbury since foundation in 597 AD (not always as an animated gif though).

This pretty theorem details of the construct of triangles using well known I/GCSE circle theorems (Exterior Angle of Triangle and Cyclic Quadrilateral).  I like the petal effect in the middle.  Never come across the word species in conjunction with triangles before.