A square ABCD is incribed in a circle. Prove that the distances between a point P on the circumference to the four vertices of the square canīt all be rational numbers.

This is just a few thoughts:

To make this easier, I put the xy origin in the center of the circle of radius R.  I put A on the positive X axis and went around counter clockwise with B,C, and D.  I put P on the circle in the first quadrant.  Because of symetry, in does not matter which quadrant.  Lets call the angle G.

Then

PA = R * (2 *(1-cosG))^.5

PB = R * (2 *(1-sinG))^.5

PC = R * (2 *(1+cosG))^.5

PD = R * (2 *(1+sinG))^.5

Now there instances where each of these can be a rational number.  But they can not all be rational for the same value of G.  I have difficulty on the proof here though.

Posted by Patrick on 2005-12-13 09:44:24