and 
in the 
plane, and you form a third point:
Suppose you are given the positions of two points
and in the plane, and you form a third point:
where 
and 
are any constants. What constraints (if
any) are there on the position of 
?
Suppose that 
; where can 
lie now?
Suppose, in addition, that 
;
what positions can 
take up now?
Finally, consider the equation
where 
. As t varies, what happens to 
?
Now we add a third point, 
; where can 
be
if
and
Now another pawl on the ratchet; we invent two new 
s with
subscripts. The first one takes the place of the old 
:
and the second relates 
and 
in the same way:
Finally 
overarches them:
Again, as t varies between 0 and 1, what does 
do?
Making---at last---the leap to an arbitrary number of points,
suppose we have n+1 of them in fixed positions, and also
n+1 scalar values 
(which we shall call
weights). As before 
and
So we form 
as
What now is the locus of 
?
Finally, another tack. Temporarily considering t and 
to
be completely unrelated variables, what is the binomial expansion of:
and of
What are they both equal to?
© Adrian Bowyer 1996