, 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
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.
, 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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