Rectangle inscribed in a circle (Find a simple criterion...)

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b2b

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Hi all - can you help me with the following question:

Let P, Q be any two points inside a give circle, C, Find a simple criterion which would allow you to decide every time whether it is possible to construct a rectangle inscribed in the given circle C (i.e. with its four vertices lying on the circle), such that P and Q lie on adjacent sides of the rectangle.

Note: If P or Q lies at a vertex of the rectangle, it is deemed to lie on each of the two sides it belongs to.
 
b2b said:
Hi all - can you help me with the following question:

Let P, Q be any two points inside a give circle, C, Find a simple criterion which would allow you to decide every time whether it is possible to construct a rectangle inscribed in the given circle C (i.e. with its four vertices lying on the circle), such that P and Q lie on adjacent sides of the rectangle.

Note: If P or Q lies at a vertex of the rectangle, it is deemed to lie on each of the two sides it belongs to.
Click to expand...
What are your thoughts?

Please share your work with us ...even if you know it is wrong.

If you are stuck at the beginning tell us and we'll start with the definitions.

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/forum/announcement.php?f=33
 
b2b said:
I tried several sketches but can't make any headway
Click to expand...
Let's start with two specific cases.

For points P and Q, do any of your sketches have one at the center and the other on the circle?

If not, draw that case.

Can you inscribe a rectangle that satisfies each of the requirements?


Do any of your sketches have both P and Q on the circle, at opposite ends of the diameter?

Consider that case.

:cool:
 
mmm4444bot said:
Let's start with two specific cases.

For points P and Q, do any of your sketches have one at the center and the other on the circle?

If not, draw that case.

Can you inscribe a rectangle that satisfies each of the requirements?


Do any of your sketches have both P and Q on the circle, at opposite ends of the diameter?

Consider that case.

:cool:
Click to expand...

One at the center and the other on the circle works.
Both on the circle works but not at opposite ends of the diameter.
 
b2b said:
One at the center and the other on the circle works.
Click to expand...
A line segment passing through the center has to touch the circle at each end, yes? How is it possible to construct the rectangle, in this case?
 
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