Perpendicular bisectors
Three Points Defining a Circle Three points uniquely define a circle. The center of a circle is the circumcenter of any triangle the circle is circumscribed about.
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- 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 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 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's equidistant to these three points
- So the way we can find it is to draw
- the perpendicular bisector of each of these sides
- and where the three perpendicular bisectors intersect
- and we've shown that they always intersect at a unique point
- that is that circumcenter.
- And I'll do that really quick 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
- and they each bisect the side,
- so 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 that,
- 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 we have three points,
- we have a unique triangle,
- that triangle has a unique circumcenter
- which is equidistant to the three points of the triangle.
- I should say three vertices of the triangle.
- and that distance between the three vertices,
- Let me draw that in a different color.
- So this distance, the length of OA,
- The length of OC, and the length of OB,
- So OA = OC = OB, which is the 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 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 you say what is the set of all the points
- on this two dimensional plane, that are equidistant
- that are a radius away from the center
- that uniquely defines a circle
- That's how we define a circle.
- And similarly, if you say,
- look, if you start with a center at O
- and you say all the points that are a circumradius
- away from O, it will uniquely identify
- a circle, and that circle will contain that 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 this
- Trying my best to draw it
- Now everything we've talked about now
- like in the last few minutes,
- is all review.
- We know all of it, 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 this clear,
- this is three non-colinear 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
- 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 a unique triangle
- with a unique circumcenter and
- unique circumradius
- 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 that
- if you give me just two points,
- there's an infinite number of triangles
- that I can 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.
- And all of these triangles are going to have
- different circumcenters and different radiuses,
- and so they are going to construct different circles
- that circumscribe about those triangles
- So for example, this would be one circle
- that could circumscribe that triangle
- You could have this circle right over here,
- so you see clearly that two points are not enough
- You need three points.
- Three points lead to a triangle, which leads to a unique circle
- So that by itself is kind of cool
- Now another question is if I have a circle
- And if it 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
- Let me draw one where this thing
- is clearly outside of the triangle.
- So we could have one that looks like this
- and clearly all three vertices sit on the circle.
- So you might 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 the circle is equidistant
- from this point.
- They're all a 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 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're going to call it point 0, I'm tired of just calling it "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 are both a radius away from the center
- So it also must sit on the perpendicular
- bisector of A and B.
- Let me draw it a little bit neater.
- So it must also be on this perpendicular bisector
- And finally, it also is equidistant from A and 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.
- And 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're intersecting at a point outside of that triangle
- And that point is the center of this circle.
- So once again, that last idea is
- O is equidistant from A and C,
- so it must sit on the perpendicular bisector of A and C
- Which would look something like this
- So once again, we see that three perpendicular bisectors
- are intersecting at a unique point
- and O really is the circumcenter
- so, if you take any circle
- and you put any triangle whose vertices
- all sit on that circle,
- the center of that circle is its circumcenter
- So we just drew a situation where this is
- a circumcenter that sits outside of the
- triangle proper.
- So point O is also going to be the
- circumcenter of this triangle.
- 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
- and we know that because it's equidistant from
- all three points of any of these triangles
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