You have an ink stamp that is so amazingly precise that, when inked and pressed down on the plane, it makes every circle whose radius is an irrational number (centered at the center of the stamp) black.
Is it possible to use the stamp three times and make every point in the plane black?
If it is possible, where would you center the three stamps?
(In reply to
Solution by Brian Smith)
{Let x be any positive irrational number.
Let AC be a line segment with length 2x.
Let B be the midpoint of AC.
Let P be any point in the plane.
Let u be the length of PA.
Let v be the length of PB
Let w be the length of PC.
Let T be the measure of angle ABP.}
Further refinement is required unless I've missed a constraint of your solution.
Let x=any positive irrational number (okay x=root3)
Let p=any point on the plane (okay choose p so that t=90 degrees and v=1)
Geomety works the following lengths;
u=w=2
v=1
t=90,pi
This causes your equation to produce no contradiction, ie.,
v^2 = (u^2 + w^2 - 2x^2)/2
1^2 = (2^2 + 2^2 - 2(√3)^2)/2
1 = (4 + 4 - 6) /2
1=1
all with x->irrational and u,v,w -> rational
re: Solution
