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# Geometric constructions: circle-inscribed equilateral triangle

## Video transcript

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 looks pretty pretty good and now let's think about something if I were to draw this segment right over here this of course it 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 centered at our new circle and take it out here now this is equal to the radius of the new circle which is the radius same as the radius of the old circle is going to be the same as this length here so these two segments have the same length now if I were to construct if I were to connect at 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 and equilateral triangle now why is 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 why is that why is this being 60 degrees interesting well imagine if we constructed another triangle out here just symmetrically but kind of flip 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 the triangle since that's a third of the way around the triangle if I were to connect if I were to connect these two dots if I were to connect these two dots that is going to be this this right over here is going to be a side of our equilateral triangle if this right over here it's it's it's secant to an arc that is one third of the entire circle and now I can keep doing this let's move 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 circumference or chol and what I 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 this right this angle that I haven't fully drawn or this arc you could say is 120 degrees so this this is going to be one side of our equilateral triangle it's secant - 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 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