Constructing regular polygons inscribed in circles
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Construct a square inscribed inside the circle. And in order to do this, we just have to remember that a square, what we know of a square is all four sides are congruent and they intersect at right angles. And we also have to remember that the two diagonals of the square are going to be perpendicular bisectors of each other. So let's see if we can construct two lines that are perpendicular bisectors of each other. And essentially, where those two lines intersect our bigger circle, those are going to be the vertices of our square. So let's throw a straight edge right over here. And let's make a diameter. So that's a diameter right over here. It just goes through the circle, goes through the center of the circle, to two sides of the circle. And now, let's think about how we can construct a perpendicular bisector of this. And we've done this in other compass construction or construction videos. But what we can do is we can put a circle-- let's throw a circle right over here. We've got to make its radius bigger than the center. And what we're going to do is we're going to reuse this. We're going to make another circle that's the exact same size. Put it there. And where they intersect is going to be exactly along-- those two points of intersection are going to be along a perpendicular bisector. So that's one of them. Let's do another one. I want a circle of the exact same dimensions. So I'll center it at the same place. I'll drag it out there. That looks pretty good. I'll move it on to this side, the other side of my diameter. So that looks pretty good. And notice, if I connect that point to that point, I will have constructed a perpendicular bisector of this original segment. So let's do that. Let's connect those two points. So that point and that point. And then, we could just keep going all the way to the end of the circle. Go all the way over there. That looks pretty good. And now, we just have to connect these four points to have a square. So let's do that. So I'll connect to that and that. And then I will connect, throw another straight edge there. I will connect that with that. And then, two more to go. I'll connect this with that, and then one more. I can connect this with that, and there you go. I have a shape whose vertices intersect the circle. And its diagonals, this diagonal and this diagonal, these are perpendicular bisectors.