Copyright © University of Cambridge. All rights reserved.
'Shape and Territory' printed from http://nrich.maths.org/
Why do this problem?
It provides practice in using the tan formula and an opportunity to
reflect on the bigger picture beyond school mathematics.
The result can be proved without using the fact that the angles add
up to 180 degrees and this could be taken up as a challenge.
Then ask "What is the 'territory' this result belongs to?"
Sue's proof given here uses the fact that the angles of the
triangle $ABC$ add up to 180 degrees. However it is just as easy to
prove this result without using the fact about the sum of the
angles of the triangle so it must be true for triangles 'living in
other geometries' where the angles of triangles do not add up to
180 degrees such as Spherical Geometry.
The lines in Spherical Geometry are great circles on the surface of
the sphere. By drawing lines like the lines of longitude and the
equator on the earth you will soon be able to convince yourself
that spherical triangles have angle sums greater than 180 degrees.
This expression gives a certain property for triangles for all 3
geometries, for Euclidean Geometry where the angles of triangles
add up to 180 degrees, for Spherical (also called Elliptical)
Geometry where the angles of triangles add up to more than 180
degrees and for Hyperbolic Geometry where the angles of triangles
add up to less than 180 degrees.
Here we have $(A-B)$, $(B-C)$ and $(C-A)$. How can we write this
using only 2 variables?
See the articles Strange
which is accompanied by a lot of ideas for project
work suitable for 12 year olds and older students on Non-Euclidean
See also the articles
How Many Geometries Are There?
When the Angles of a Triangle don't add up to 180 degrees.