Geometric constructions: circle-inscribed regular hexagon

construct a regular hexagon inscribed inside the circle so I'm going to do first I'm going to draw a diameter of the circle actually I'm going to go beyond the diameter of the circle I'm just going to draw a line that goes through the center of the circle and just keeps on going and I'm going to make it flat so it goes directly through so this one right over here goes directly through the center and now what I'm going to do is I'm going to construct a circle that's the exact same dimensions of this circle that's already been drawn so let me put this one right over here let me make it the same dimensions and now what I'm going to do is I'm going to move it over so that this new circle intersects the center of the old circle and these circles are the same size so notice this Center intersects the old circle and the new circle itself intersects the center of the old circle now the reason why this is interesting we already know that this distance the distance between these two centers this is equal to a radius we also know we also know me at a straightedge we also know that this distance right over here is equal to a radius it's equal to the radius of our new circle right over here we also know that this distance right over here is equal to a radius of our old circle and that they both have the same radius so this is a radius between these two points between these two points in radius and then in between these two points is a radius so now I have constructed an equilateral triangle and essentially I have to just do this six times and then I'm going to have a hexagon inscribed inside the circle let me do it again so go from here to here this is a radius of my new circle which is the same as the radius of the old circle and I could go from here to here that's the radius of my old circle so I have another equilateral triangle radius radius radius another equilateral triangle I just kind of do this I have to do this four more times so let me go to my original let's see let me make sure I can well it's actually going to be hard for me to let me just add another circle here to do it on the other side so if I put it the center of it right I want to move it a little bit right over I want to make it the same size so that's close enough and let's see that looks pretty pretty close that's the same size and let me move it over over here now I want this Center to be on the circle right like that and now I'm ready to draw some more equilateral triangles so a day and really I don't have to even draw the inside of it now I see my my six vertices for my hexagon here here here here here here and I think you're satisfied now that it you could break this up into six equal lateral triangles so let's do that so this would be the base of one of those equilateral triangles and actually let me move these let me move this one out of the way I can move this one right over here because I really just care about the hexagon itself and I can move this right over here but we know that these are all the lengths of the radius anyway actually I'm not even having to change the length there then I have to just connect one more right down here so let me add another straight edge connect those two points and I would have done it I would have constructed my I've constructed my regular hexagon inscribed in the circle