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# Three points defining a circle

## Video transcript

we know that three points define a triangle so if I were to take three random points here so let's call that point a point B and then let's say this is point C right over here if we say that these three points are the vertices of a triangle they define a unique triangle so this would be triangle a try to draw my lines as straight as possible triangle a b c now we've also learned in the last few videos that triangle ABC has a unique circumcenter and that is a point that is equidistant to these three vertices it's equidistant to these three points so and we the way we can find it is we draw a perpendicular bisector of each of these sides and where the three perpendicular bisectors intersect and we show that they always intersect at a unique point that is that circumcenter and I'll do it really quick right over here so let's say this is the perpendicular bisector of that side this is the perpendicular bisector of that side and this is the perpendicular bisector I want to go from this is the perpendicular bisector of that side so these are all perpendicular this is perpendicular and they each bisect the sides so this side B to this point is going to be equal to this point to a a to this point is going to be equal to that point to see see that this point is going to be equal to that point to B and this point right over here we've already talked about we'll call that point O we call that the circumcenter o is the circumcenter circumcenter this is all a little bit of review so if you have three points you have a unique triangle that unique triangle has a unique circumcenter which is equidistant to the three points of the triangle three I should say the three vertices of the triangle and that distance that distance between the circumcenter and the three points the three vertices I should say so let me draw that in a different color so this distance Oh a the length of O a the length of the length of O see and the length of OB so o a equal to OC is equal to OB which is the circumradius circum radius and it we've learned when we first talked about circles if you give me a point and if we find the locus of all points that are equidistant from that point then that is a circle i when i say locus all i just means the set of all points if you give me any point if you give me any point right over here so that's an arbitrary point and you also specify a radius and you say what is the set of all the points on this two-dimensional plane that are equidistant for that are that radius away from the center it uniquely defines a circle that's how we defined a circle right over here and similarly if you say look if you start with the center at O and you say all of the points that are the circumradius away from o it will uniquely identify a circle and that circle will contain the points a B and C because those are those are the circum radius away from O so they are included in that set so the circle would look something like let me draw it it would look something like this dry my best two best to draw it just like that now everything we talked about just now in the last few minutes is all a review we know all of this but I went over it just to kind of reinstate a pretty interesting idea that if you give me three points if you give me three points that defines a unique triangle unique triangle and if you have a unique triangle and let me let me make it clear this is three this is three non collinear points so three points not on the same line the same line if you have three points that are not on the same line that defines a unique triangle for any unique triangle you have a unique circumcenter unique circumcenter circum Center and circum radius all right I'll rewrite it I'm gonna get lazy confuse you circum radius and if you give me any point in space any unique point and radius the set of all points that are exactly that radius away from it that defines a unique circle that defines a unique a unique circle so we went through all of this business of talking about the unique triangle and unique circumcenter and the unique radius to really just show you that if you give me any three points if you give me any three points that eventually really just defines a unique circle so just as you need to three points to define a triangle you also need three points to define a circle two points won't do it and one way to think about it is if you give me two points if you give me two points there's there's an infinite number of triangles that I construct with those two points because I can put the third point anywhere I could construct this triangle I could construct this triangle I could construct this triangle I can construct this triangle and all of these triangles are going to have different circumcenter 's and different radiuses and so they're gonna have different circles that circle that circumscribe about those triangles so this one so for example this would be one circle that could go around that could subscribe that triangle you could have this circle right over here so you see clearly very clearly that two points are not enough you need three points three points lead to a triangle you didn't lead to a unique circle so that by itself is kind of cool now another question is is if I have just a circle and if it's circumcircle right if it's circumscribed about an arbitrary triangle is the center of that is the center of that circle necessarily the circumcenter so let's think about that a little bit because there are some there are some I guess unnatural or I would say unnatural non-intuitive cases here so if I draw a circle if I draw a circle right over here its center is right over there and if I draw an arbitrary triangle where all of the vertices of that triangle are on are on this circle is this Center necessarily the circumcenter of that triangle so let me draw let me draw a crazy situation so let me draw one where this this thing is clearly outside of the triangle so that we could have a triangle that looks like this and it's clearly all three vertices it on the circle so you might at first say wait there's no way this could be the circumcenter it's not even inside the triangle but remember this point this point right here is equidistant to every point on the circle I should say every point on this circle is equidistant from this point they are all the radius away and all three points of this triangle are on the circle so they are all exactly Arabia severe so this distance right over here is going to be a radius this distance right over here is going to be a radius and this distance right over here is going to be a radius now this point is clearly equidistant from that point and that point we know that it's exactly R away from both of those both of those vertices of the triangle so if it's equidistant to reprove this in a previous video if it's equidistant from both of those points it must be on the perpendicular bisector of the segment that joins those two points so this must be on the perpendicular bisector so that's it's perpendicular and it bisects that segment right over there but we can make the same argument for this segment right over here because this point is R from this the center we're call it oh I'm tired of just saying this point point O is equidistant from let me label these so let's call this a B C so we already said point O is equidistant from C and B so it must be on the perpendicular bisector of BC and it's also equidistant from a and B it's R away from both because a and B both sit on the circle they're both a radius away from the center so it also must sit on the perpendicular bisector of a B so it also must sit on the perpendicular that doesn't look as that didn't let me draw it a little bit neater there you go so it must also be on this perpendicular bisector and then finally it also is equidistant from a it's for the same distance from a as it is from C because those are both R away they both sit on the circle so it must be on the perpendicular bisector of AC as well of AC as well so AC is right over here and this is what the interesting thing is we're seeing that the interest that need the three the three perpendicular bisectors of the three sides of this triangle they do definitely intersect but they are intersecting at a point outside of that circle outside of that triangle and that point is the center of this circle so once again the last point right that the last idea is oh is equidistant from a and C so it must sit on the perpendicular bisector of AC which would look something like this which would look something like something like that so once again we see the three perpendicular bisectors are intersecting at a unique point and oh really is the circumcenter so if you take any circle if you take a circle if you put any triangle whose vertices sit on the circle the center of that circle is its circumcenter so we just drew a situation where this is the circumcenter that sits outside of the triangle proper so point O is also going to be the circumcenter of of this triangle of this triangle right over here and point O is also going to be the circumcenter of this triangle right over here it's going to sit on all three perpendicular bisectors of this and we know that because it's equidistant from all three points of of any of these triangles that where the vertices sit on the circle itself