



There
are a couple of tricky things



about
angles and circles that you should know. 





Say you
have an angle 


that
comes from the center of a circle. 


Like
this: 











And
another angle that comes from the far side of the circle 


to the
same points, like this: 











The trick
is that the angle that starts at the edge 


is
exactly half as big as the angle that starts in the middle! 





Example: 











Well,
maybe that one wasn't that big a surprise. 


But I bet
this next part is ... 





The angle
that starts from the edge 


doesn't
have to start exactly on the other side of the circle. 


It can
start ANYWHERE outside the two points! 











That is
one of those amazing but true things in math. 


When I
first heard about it (about a million years ago) 


I didn't
believe it. 


But it's
true. 





The next
thing we are going to talk about 


may be
even more amazing. 





First
some info ... 








Now back
to the incredible part. 





One of
these "from the edge" angles 


could be
drawn like this: 





Start
with the big angle from the center ... 











Draw a
line like this ... 











and a
tangent line, like this ... 











The angle
that this makes ALSO counts 


as one of
the half as big angles! 





Example: 





I think
that is amazing! 


Remember
that the line outside the circle MUST be tangent. 


We can't
just draw that other line in anywhere ... 











copyright 2005 Bruce Kirkpatrick 
