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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 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.