Point in a circle (geogebra question)

Question

This is a screenshot of the "Point on a circle" lesson on GeoGebra:

I have to answer the lesson's first 3 question (shown below), and I could use some help. I tried for 3 days, and I don't seem to get it.

1. What do you notice? Any conjecture(s)? Proof(s)?
2. Can the diagonals of the four squares form a quadrilateral? If yes, does the order of the colored diagonals matter? Why or why not?
3. Under what circumstances can the diagonals of the four squares not form a quadrilateral?

• In the first question, I answered that I noticed that the ratio between the area of the circle and the area of the four squares is a constant $ \frac{\pi}{2}$. I'm still stuck in how to prove it, if anyone can give me a hint on how to start it will be great!

• In the second question: According to the third question, the answer is "yes, the diagonals can form a quadrilateral", but I don't see why. A quadrilateral has 4 edges, each two has a joint point, and here the only joint point is the joint point of all 4 squares, and it's impossible to make a quadrilateral? When i try, there is only 1 way for the diagonals to make a quadrilateral, and that is if the joint point of the 4 squares is in the middle of the circle and they are in the same size.

• And of course I cannot solve the third question because it is based on the second.

If your read it until here THANK YOU! I will be really thankful for who can help!

Answer 1

You seem to have made all the correct observations already. You just seem to need some confirmation that your observations are indeed correct. So here are my answers to the questions:

1. From just the screenshot I would guess that the total area of the four squares is twice the radius squared. This is equivalent to your observation.
2. If the point is in the center of the circle, then all four squares are the same size, and they line up to form a larger square. The diagonals that then form a diamond (in fact a tilted square), which is a quadrilateral.
3. From the answer to the previous question, it is not hard to see that for the diagonals to form a quadrilateral, each pair of adjacent squares must line up. This happens for all four squares at once if and only if the point is at the center.

Of course you may want to go into more detail than I just did, but this is the gist of it.