Three points defining a circle

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 ABC. 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 equidistant to these three points. So the way can 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 that this is the perpendicular bisector of that side, this is the perpendicular bisector of that side, and this is the perpendicular bisector of that side. So these are all perpendicular, this is perpendicular, and they each bisect the sides. 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 C. C to 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. 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 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, OA, the length of OA, the length of OC, and the length of OB, so OA is equal to OC is equal to OB, which is the c circumradius. And 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. And when I say a locus, all I mean is, the set of all points. If you give me any point right over here, so that's an arbitrary point, and you also specify a radius, and say what is the set of all the points on this two dimensional plane that are equidistant, 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 the circumradius 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-- trying my best to draw it, just like that. Everything we've talked about, just now within the last few minutes, is all 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 that defines a unique triangle, and if you have a unique triangle-- And let me make it clear. This is three non-collinear points, so three points not on 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 and circumradius. I'll rewrite it, I don't want to get lazy and confuse you-- circumradius. And if you give me any point in space, any unique point, and a radius, the set of all points that are exactly that radius away from it, that defines a unique circle. So we went through all of this business of talking about the unique triangle, and the unique circumcenter, and the unique radius, to really just show you that if you give me any three points that eventually, really, just defines a unique circle. So just as you need 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, there's an infinite number of triangles that I construct with those two points, because I could 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 circumcenters and different radiuses. And so they're going to have different circles that circumscribe about those triangles. So this one-- so for example, this would be one circle that could go around, that could circumscribe 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, lead to a unique circle. So that by itself is kind of cool. Now, another question is, if I have just a circle, and if it's circumscribed about an arbitrary triangle, is the center of that circle necessarily the circumcenter? So let's think about that a little bit, because there are some non-intuitive cases here. So 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 this circle, is this center necessarily the circumcenter of that triangle? So let me draw a crazy situation. So let me draw one where 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 sit 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 right here is equidistant to every point on the circle. I should say, every point on this circle is equidistant from this point, they're all the radius away. And all three points of this triangle are on the circle, so they are all exactly a radius away from this point right over here. 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 vertices of the triangle. So if it's equidistant-- and we proved 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 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 the center-- we'll call it O, I am 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 AB. 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 is 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. So AC is right over here. This is what the interesting thing is, we're seeing that 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 triangle. And that point is the center of the circle. So once again, the last idea is, O 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 that. So once again, we see the three perpendicular bisectors are intersecting at a unique point, and O really is the circumcenter. So if you take any circle, if you take a circle, and 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 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 any of these triangles where the vertices sit on the circle itself.