equable triangles

the proof that there are only five equable triangles (integer sided) was done in 1904

this version of a proof is largely due to David Wells with small bits of help from me

it is complex but involves only GCSE tools

roof truss

one roof making company calls the various truss designs:
Howe truss
Fink truss
Attic truss

which one uses most wood?

quite long and complex calculations involving the sine and cosine rule

symmetry is assumed

the Fink truss does not have struts that are perpendicular to the main, 6m lengths (even though it looks like they might be)

cosine rule

a powerpoint is here

the ambiguous case, where you are given angle, side, side (condition: A.S.S.) can lead to two integer solutions, obtained by factorising and solving a quadratic equation

area 5

not using right angled triangles and Pythagoras...
using:

• the cosine rule
• normal trigonometry
• area of a triangle, A = 1/2 ab sin C

cosine rule from a picture

establish that the yellow parallelograms have an area of abcosP
also establish that the overall shapes are congruent

hence to the cosine rule...

[ from Wikipedia ]