Geometric constructions: circle-inscribed equilateral triangle

Construct an equilateral triangle inscribed inside the circle. So let me construct a circle that has the exact same dimensions as our original circle. Looks pretty good. And now, let me move this center, so it sits on our original circle. So they now sit on each other. Or their centers now sit on each other. So I can make it, and that looks pretty good. And now, let's think about something. If I were to draw this segment right over here, this, of course, has the length of the radius. Now, let's do another one. And that's either of their radii, because they have the same length. Now, let's just center this at our new circle and take it out here. Now, this is equal to the radius of the new circle, which is the same as the radius of the old circle. It's going to be the same as this length here. So these two segments have the same length. Now, if I were to connect that point to that point, this is a radius of our original circle. And so it's going to have the same length as these two. So this right over here, I have constructed an equilateral triangle. Now, why is this at all useful? Well, we know that the angles in an equilateral triangle are 60 degrees. So we know that this angle right over here is 60 degrees. Now, why is this being 60 degrees interesting? Well, imagine if we constructed another triangle out here, just symmetrically, but kind of flipped down just like this. Well, the same argument, this angle right over here between these two edges, this is also going to be 60 degrees. So this entire interior angle, if we add those two up, are going to be 120 degrees. Now, why is that interesting? Well, if this interior angle is 120 degrees, then that means that this arc right over here is 120 degrees. Or it's a third of the way around the triangle. This right over here is a third of the way around the triangle. Since that's a third of the way around the triangle, if I were to connect these two dots, that is going to be, this right over here is going to be a side of our equilateral triangle. This right over here, it's secant to an arc that is 1/3 of the entire circle. And now, I can keep doing this. Let's move-- I'll reuse these-- let's move our circle around. And so now, I'm going to move my circle along the circle. And once again, I just want to intersect these two points. And so now, let's see, I could take one of these, take it there, take it there, same exact argument. This angle that I haven't fully drawn, or this arc you could say, is 120 degrees. So this is going to be one side of our equilateral triangle. It's secant to an arc of 120 degrees. So let's move this around again. Actually, we don't even have to move this around anymore. We could just connect those last two dots. So we could just connect this one. Actually, I just want to, let's connect that one to that one. And just like that, and we're done. We have constructed our equilateral triangle.