We can solve a quadratic equation of the form
x2 -
sx +
p = 0,
, using the standard construction tools of compass and straightedge. The method has been attributed to critic Thomas Carlyle.
Construct the circle in the Cartesian plane with center
 "\left ( \frac{s}{2},\frac{p+1}{2} \right )")
and passing through
A(0,1). By symmetry, the circle also passes through
and
.
Because the circle has center
and passes through
A(0,1), the equation of the circle is
This reduces to
x2 - sx + y2 - (p + 1)y + p = 0
So the
x-intercepts of the circle are the solutions to
x2 -
sx +
p = 0.
Alternate justification:
The segment joining the
x-intercepts has a length
, hence
x1 +
x2 =
s.
intercepts the arc
AX1X2, and
intercepts the opposite arc, hence the two angles are supplementary. But
is also supplementary with
, so
is congruent to
, which in turn impies that
. Hence
,
which implies that
, so
x1 x2 =
p.
Thus (
x -
x1)(
x -
x2) =
x2 -
sx +
p, and the solutions to
x2 -
sx +
p = 0 are
x1 and
x2 .
3 comments:
you might want to change the y coordinate of your circle center -- shows as \frac{p-1}{2} when you want \frac{p+1}{2}
you may want to change the y-coordinate of your circle center -- it reads \frac{p-1}{2} when you want \frac{p+1}{2}
Thanks.
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