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### Course: Geometry (all content)>Unit 4

Lesson 4: Perpendicular bisectors

# Three points defining a circle

Three points uniquely define a circle. If you circumscribe a circle around a triangle, the circumcenter of that triangle will also be the center of that circle. Created by Sal Khan.

## Want to join the conversation?

• Could someone explain to me how you can have 'o' as the circumcenter of a triangle it is not in? ( at about the triangle i am confused about is drawn.)
• That is because the circumcenter doesn't have to be inside the triangle in all cases. In fact, in acute triangles it is always inside the triangle; in right triangles, it is always on the triangle, and in obtuse triangles, the circumcenter is always outside the triangle!
• At , why does Sal write that any unique triangle has a unique circumcenter and circumradius?

I do agree that 3 points define a unique triangle, but different triangles can have the same circumcenter and circumradius, right? It is proven by the fact that you can draw multiple triangles inside a circle where the vertices of the triangle are on the circle.

Therefore a triangle doesn't have a unique circumcenter and circumradius, correct?
• Okay. Your conclusion is not correct, but let me illustrate why:

(1) "Therefore a triangle doesn't have a unique circumcenter and circumradius."
This is false. This statement, however, is true:
(2) Therefore a given circle doesn't have a unique inscribed triangle.

Sal demonstrates that three noncollinear points uniquely determine a circle. You have correcly argued than the converse is not true. That is,
A given circle does not uniquely determine three noncollinear points.

The only mistake is your conclusion. To get a clear picture of how 'unique' can go one direction and not the other, I use parents and children. I have one and only one biological mother. My mother does not have one and only one child (I have a sister).

However, when Sal says at "all of these triangles are going to have different circumcenters," that is not true, (or just ambiguous depending on what he means). Given two points, using a third point we can construct an infinite number of triangles that all lie on different circles; however, we can also use a third point to define an infinite number of triangles on the same circle.

Is that the error you're trying to pinpoint?

Be well and have fun.
• Would two points work for defining a circle if they indicated the length of the diameter?
• Actually, all of the points are assumed to be on the circumference, so by standard convention, you need at least three points.
• What are or what is Euclid's elements?
• so it's NOT the periodic table of elements because I was thinking about that when you said "Elements."
• What does arbituary mean?
• Arbitrary in this video sort of means 'random.' It means the triangle wasn't chosen for any special reason- he could have defined any other triangle.
• Re: . Is Sal saying that all (emphasis) 3 points cannot be on the same line to have a triangle, though it is true that there will always be 2 of the 3 points which are on the same line.
• Any two points, no matter how far apart or weirdly placed (even in 3D, if the two points were at any depth), will always form a line.
But if you get three points on the same line, all it would form is a line, and not a triangle, which is a polygon and not a line.
To better explain, if you had three points that fell on a same line, you couldn't make a triangle out of them, consequently preventing you from taking a circumcenter (because you don't have a triangle), and finally not letting you make a circle.
• Regarding circumcentre, centroid and orthocentre, could someone please tell me more about the Euler line in a triangle?(I know this is not about circles, I am sorry.)
• so it is just the middle of the circle for both ones