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Here we have a square and 2 points E and F on the sides of the square and a circle whose
diameter is those 2 points. Notice the circle goes through A. Does it always go through
A. Move B. Let us move A. Now let us move the endpoints the diameter. It always goes
through A. What is happenging here? Why does the circle always go through A.
Now here is how to set up this GeoGebra Explorer. What we need is a right angle. We are going
to draw a square, but you only really need a right angle. We draw a rectangle also, but
we are going to draw a square. So go get our Regular polygon tool. Remember never click
on (0,0) because then the square is tied to (0,0). So let us click here on (1,1) and then
on (4,1) and how many sides? We want 4 because we want it to be a square. So we click on
Okay. And there is a square. What else do we need. We need a point on the
base of the square. Get the New point tool and click anywhere on the base. We want it
on the segment. And, click on the side. So now we have 2 points on the base and on
the side of the square. What we want to do is draw a circle with this diameter EF. In
order to draw a circle, we need a line segment. So take our segment tool from E to F.We need
a center of the circle so let us quickly get a midpoint. And we want the circle with center
G and point E (EF is the diameter). Look it is going through A. Does that always
happen? Let us move our square. Look it always is going through A. If we move A, it goes
through A. If I move these points (E and F), it always goes through A.
What is happening here? Why does this circle always go through A? Well I looked at it and
I thought "This reminds me of Thales Theorem". Now don't be afraid. You know what Thales
Theorem is. But go and look it up on Wikipedia and see if you can find out why this circle
always goes through A.