Determine if point lies on inscribed square

[ged25]

How do you determine if a point lies on the sides of a square inscribed on a circle ?
I'm given only the radius of the circle and the circle is centered at the origin.
My idea was to find the equation for each side and then to test the given point.
However, I just realized that you can inscribe a square in many different ways.
I'm stumped on how to do this.

 

[micromass]

Do you mean "determine if a point lies on A certain square" or on a specific square? If you need to prove the last, then you really need some more information.

The first question is a bit more fun. Try to see it intuitively first. Draw an inscribed square and mentally change the square to another inscribed square. You will see that you get a second (smaller) circle. Points lying inside the circle do not belong to a inscribed square, and points lying inbetween the cricles do belong to a insribed square.
The only thing you need to do now is to find the radius of the circle.

I'm sorry if you understood nothing of what I tried to describe, but it's hard to explain. I suggest download the free program geogebra and play a bit with it. You'll see very soon what I mean...

 

[ged25]

Its the first question.
I do not understand how you can get a smaller circle when you shift the square. Maybe I'm doing it wrong but I thought that when you shift the square, the end points of the square move along the circle. The diagonals of the square will be equal to the diameter of the circle and will remain the same regardless of the square is drawn.

 

[micromass]

You don't shift the square, but you rotate the square. But perhaps you meant that.
It's true that the endpoints of the square will still lie on the circle, but the sides of the square will create an inscribed circle. The side of every inscribed square will be a tangent line to this inscribed circle. This fact would make it easy to find an equation for this circle...

 

[ged25]

Ok, I get it now. I seem to vaguely remember an animation that showed this.