Carlyle Circle
Consider a quadratic equation 
where
and 
denote signed
lengths. The circle which has the points 
and 
as a diameter
is then called the Carlyle circle 
of the equation.
The center of 
is then
at the midpoint of 
, 
,
which is also the midpoint of 
and 
. Call the points at which 
crosses
the x-axis 
and
(with 
). Then
 |
(1)
|
 |
(2)
|
 |
(3)
|
so 
and 
are the roots
of the quadratic equation.
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