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.
