Talk:Circumcircle

Incomplete information
The session "Cyclic n-gons" seems to be incomplete. The last sentence is cut off in the middle after the word "alte", which doesn't make any sense at all. Please help correct it. Thanks. User:Yuen.teddy —Preceding undated comment added 13:13, 5 August 2011 (UTC).

Discussion: Merge with circumcircle
I assert that circumscribed circle and circumcircle not be merged. They are fundamentally different concepts with different definitions, and to merge them would show a profound lack of geometric sophistication. For example, if you draw a five-point star, this star has no circumscribed circle (since the star has 10 vertices), but it does indeed have a circumcircle. The circumcircle does not have to touch every vertex; instead, it merely has to be the unique circle with the smallest radius such that the entirety of the star's interior is bounded within it. User:KyleGoetz

Triangles: Circumscribed circle is a circumcircle
A circumscribed circle for any triangle is the same circle as the circumcenter. Whoever wrote otherwise is absolutely wrong: http://mathworld.wolfram.com/Circumcircle.html User:KyleGoetz Just to clarify this with my comment above concerning the differences between a circumcircle and a circumscribed circle: for a triangle, they are the same; however, for other polygons this may not be true. User:KyleGoetz

I think it is very wrong to have a definition for triangles that is incompatible with the definition for polygons in general, but I also don't think the circumscribed circle and circumcircle are ever different. At best note the "smallest enclosing circle" version in passing as a variant usage rather than making it the main definition of the word. As source I cite this paper — it's my own work, so I'm not going to edit into the actual article, but it's a well respected survey from a number of years ago and clearly states (top of page 9, and theorem 1 page 13) that circumcircle = circumscribing circle for triangles, and that the smallest enclosing circle is something different. The only single word synonym I've seen in professional usage for smallest enclosing circle is minidisk or minidisc (or miniball in higher dimensions). —David Eppstein 04:49, 15 October 2006 (UTC)


 * Yeah, there is no difference between circumscribed circle and a circumcircle. Any polygon has a circumcircle, and when polygon's vertices happen to be all on that circumcircle, it is called a circumscribed circle. Oleg Alexandrov (talk) 05:05, 15 October 2006 (UTC)


 * Er. There are polygons which have all vertices on a circle, but for which that circle is not the smallest enclosing circle. Polygons other than triangles, even. So I'm not sure which circle your comment is referring to. To me, "circumscribed circle" and "circumcircle" can only mean a circle passing through all vertices, even if that circle is not the smallest one enclosing the polygon. Anything else needs a different name. —David Eppstein 05:17, 15 October 2006 (UTC)
 * OK, then my changes to the article were wrong. I still think that my merger of circumcircle into circumscribed circle was a good thing. I will try to fix my errors tomorrow. Oleg Alexandrov (talk) 05:34, 15 October 2006 (UTC)
 * Thanks for fixing my mistake! Oleg Alexandrov (talk) 16:03, 15 October 2006 (UTC)


 * David, I agree with you, and I did not mean to say anything different than what you did. It's now been so long since I edited this article, that I cannot remember exactly what brought about my mentioning of triangles specifically in the talk page, but I think it's because the article itself explicitly said that circumcenter and circumscribed circle were the same thing, and used a triangle as proof of the statement. I merely pointed out this was an erroneous proof that the two concepts are the same. KyleGoetz 23:25, 17 December 2006 (UTC)


 * Is the property that all triangles can be circumscribed by a circle a defining property for triangles? What happens to a degenerate (colinear) triangle if that is the case? — Preceding unsigned comment added by Knservis (talk • contribs) 08:48, 2 May 2014 (UTC)
 * In that case the circumscribing circle degenerates to the line through the three points. —David Eppstein (talk) 15:19, 2 May 2014 (UTC)

Incorrect Formula?
is the formula under: "Coordinates of circumcenter" right, i'm not sure philb 14:53, 24 March 2007 (UTC)

The formula is correct. Note that these barycentric coordinates are not "normalized" (as in areal coordinates, where λ1+λ2+λ3=1). If you want the barycentric coordinates that are most commonly used, divide the entire thing by the sum of the components 128.237.234.118 (talk) 19:37, 13 April 2008 (UTC) I don't think this is right. If you normalize the barycentric coordinates, the center will always be located inside the triangle, which is not correct. Also, barycentric coordinates of the circumcenter should be invariant with respect to scaling, which these are not. 131.246.191.182 (talk) 13:01, 22 January 2009 (UTC)
 * What makes you think the given center will always be located inside the triangle? You have
 * $$ a^2 + b^2 - c^2 \, $$
 * and that is negative in case the triangle has an obtuse angle between the sides of lengths a and b. A negative barycentric coordinate entails a point outside of the triangle, just as one would expect for an obtuse triangle. Michael Hardy (talk) 22:34, 22 January 2009 (UTC)
 * ...and now I've checked this and the given coordinates are correct. Michael Hardy (talk) 18:35, 31 January 2009 (UTC)

Frustrating Merges
I agree with KyleGoetz. I spent time separating circumsphere and circumscribed sphere from eachother just to have someone come along and merge them. One must "touch 3 points and enclose the polygon" and the other "touch all the vertices of the polygon". As a software engineer who has worked on rigid body dynamics, there is a need for the former (circumscribed sphere) to define the collision area of ANY arbitrary polyhedron. Circumspheres (and circles) cannot be applied to any arbitrary polyhedra (or polygons) make the need for distincition clear. Coder0xff 19:24, 29 April 2007 (UTC)


 * The article as it stands correctly distinguishes between two concepts, (1) touching all vertices, and (2) enclosing and smallest. As I understand it you are trying to introduce a distinction between these two and a third concept, (3) touching three vertices and enclosing. Circles satisfying this third definition always exist but are uniquely defined only when the convex hull of the polygon is cyclic. It's a reasonable definition, though I think not as useful as the other two. But can you provide reliable sources in the mathematical literature clearly stating that definition (3) is associated with one of the phrases "circumcircle" or "circumscribed circle" and that definition (1) is associated with the other phrase? Because I don't think I've ever seen such a definition in the papers I've read. This is not the place for making up distinctions that do not represent what is already in the literature. —David Eppstein 19:33, 29 April 2007 (UTC)

Formula for calculating the length A
If I start out with the radius of a circle first, and inscribe an equilateral triangle inside the circle, how would I go about calculating the length A of any of the triangle's sides? If I had a notepad here, why, I guess I'd go ahead and figure it out myself, but my desk really doesn't have any room for that kind of stuff.

In any case, the main article contains a formula that lets you calculate the circumference of the circumscribed circle, if you start out with any of the sides of an equilateral triangle, but the article could be improved by including a way of figuring out the length of any of the triangle's sides, if you start out with a circle first. It makes a small difference which you start out with first, but one formula might be more easily understood than the other. 216.99.219.73 (talk) 01:07, 19 June 2009 (UTC)


 * If you start out with the length of the Radius of the circumscribing circle, R, you can calculate the length of any given side of the Triangle, A, by using the following formula:


 * A = 3*R / square root of 3     Hope that helps.   216.99.198.254 (talk) 05:26, 21 June 2009 (UTC)

Polar Coordinates
The main article uses Cartesian coordinates when it describes a circumcircle around an equilateral triangle. Are there any elegant equations when using Polar coordinates instead? Are the formulas any less elegant when the triangle is no longer equilateral? 216.99.219.73 (talk) 01:07, 19 June 2009 (UTC)

Coordinates of circumcenter
Formula of circumcenter O for triangle AB'C' is given. Should it be mentioned that for triangle ABC circumcenter would be O + A ? Lost couple of hours figuring this out. Yatagarasu42 (talk) 12:48, 18 May 2010 (UTC)

More Detail Please
The leap from the system of four equations for the vertices and an arbitrary point to the determinate form is for me too far to take without explanation. I don't recall enough linear algebra to follow the reductions. Can someone show the steps from the four equations for the lengths of the differences of the vectors being equal to the length of the radius squared, through the polarization identity transformation, to the matrix form, and then to the determinate expression?

It is the derivation of this matrix which, for the moment, is beyond me:
 * $$\begin{vmatrix}

\end{vmatrix}$$ Lonniev (talk) 14:33, 3 June 2011 (UTC)
 * \mathbf{v}|^2 & -2v_x & -2v_y & -1 \\
 * \mathbf{A}|^2 & -2A_x & -2A_y & -1 \\
 * \mathbf{B}|^2 & -2B_x & -2B_y & -1 \\
 * \mathbf{C}|^2 & -2C_x & -2C_y & -1

I support this. Which one of polarization identities is used? How is kernel related here? — Preceding unsigned comment added by 46.146.133.212 (talk • contribs)  7 July 2013

Lets see have a go
 * $$|\mathbf{v}-\mathbf{u}|^2 - r^2=0$$
 * $$|\mathbf{A}-\mathbf{u}|^2 - r^2=0$$
 * $$|\mathbf{B}-\mathbf{u}|^2 - r^2=0$$
 * $$|\mathbf{C}-\mathbf{u}|^2 - r^2=0$$

expanding the first gives
 * $$(v_x-u_x)^2 + (v_y-u_y)^2-r^2=v_x^2-2 v_x u_x + v_y^2 + v_y^2 - 2 v_x u_x + u_y^2 -r^2= |\mathbf{v}|^2 - 2 v_x u_x - 2 v_y u_y + |\mathbf{u}|^2-r^2=0$$

We have similar for the other equations. We can then right the four equations as a single matrix equation
 * $$\begin{pmatrix}

\end{pmatrix} \begin{pmatrix}1 \\ u_x \\ u_y \\ r^2 - |\mathbf{u}|^2\end{pmatrix} =\begin{pmatrix}0\\0\\0\\0\end{pmatrix}$$ So multiples of $$(1, u_x, u_y, r^2 - |\mathbf{u}|^2)$$ are the kernel vectors.--Salix (talk): 13:43, 7 July 2013 (UTC)
 * \mathbf{v}|^2 & -2 v_x & -2 v_y & - 1 \\
 * \mathbf{A}|^2 & -2 A_x & -2 A_y & - 1 \\
 * \mathbf{B}|^2 & -2 B_x & -2 B_y & - 1 \\
 * \mathbf{C}|^2 & -2 C_x & -2 C_y & - 1

Circumcircle equations
This content is clearly "owned" by David Eppstein because he quickly reverts any attempt to clarify the content. Admittedly, the new content was poor mathml but it was a work in progress on existing content which was itself less than perfect. So, I ask David to belabor the content in order to get from the four equations of the squared vector differences to the determinant. David, if you prefer not to dirty the article with the details, you can explain the steps here on the discussion page. My goal is not to get my content into the article but to understand the steps in this manner of determining the coordinates of the center and the radius of the circumscribed circle. — Preceding unsigned comment added by 97.122.243.92 (talk) 03:41, 25 July 2011 (UTC)
 * Less than perfect? You had an equation with a matrix on the left hand side and a vector on the right hand side, and another equation with a matrix on the left hand side and a scalar on the right hand side. It made no sense mathematically. And I don't know what you're talking about re ownership: I've made a single edit to the article in the last two months (to undo your bad math), over which time seven other editors have been working on it. —David Eppstein (talk) 04:01, 25 July 2011 (UTC)
 * David, your "single edit" was to revert away two paragraphs while I was in the middle of editing it. Yes, the [4x4] = [1x4] was "bad math" but I about to add the column vector. I've asked for clarity a month ago on the unstated steps taken to get from the equation set to the shown determinant. No one responded. At least by attempting to modify the presentation article, someone who cares enough about the article to try to protect it has responded. Again, it is not my intent to force my content into the article, I just want to know the steps taken and I want the article to at least reference the not-shown steps if not explicitly state them. Your contribution to that goal is greatly appreciated. — Preceding unsigned comment added by 65.102.235.174 (talk) 13:44, 25 July 2011 (UTC)
 * The preview button is your friend. It shouldn't be necessary to leave the article in a bad state in the middle of a sequence of edits. —David Eppstein (talk) 16:10, 25 July 2011 (UTC)

Wrong generalization to d-dimensional points
The original article says:

Additionally, the circumcircle of a triangle embedded in d dimensions can be found using a generalized method. Let A, B, and C be d-dimensional points, which form the vertices of a triangle. We start by transposing the system to place C at the origin:
 * $$\mathbf{a} = \mathbf{A}-\mathbf{C}$$,
 * $$\mathbf{b} = \mathbf{B}-\mathbf{C}$$.

The circumradius, r, is then
 * $$r = \frac{\left\|\mathbf{a}\right\|\left\|\mathbf{b}\right\|\left\|\mathbf{a}-\mathbf{b}\right\|}

{2 \left\|\mathbf{a}\times\mathbf{b}\right\|} = \frac{\left\|\mathbf{a}-\mathbf{b}\right\|}{2 \sin\theta} = \frac{\left\|\mathbf{A}-\mathbf{B}\right\|}{2 \sin\theta}$$, where &theta; is the interior angle between a and b. The circumcenter, p0, is given by
 * $$p_0 = \frac{(\left\|\mathbf{a}\right\|^2\mathbf{b}-\left\|\mathbf{b}\right\|^2\mathbf{a})

\times (\mathbf{a} \times \mathbf{b})} {2 \left\|\mathbf{a}\times\mathbf{b}\right\|^2} + \mathbf{C}$$.

For both the computation of $$r$$ and $$p_0$$ the cross product $$\times$$ is used (this can be clearly seen in the simplification step from $$r$$). However, the cross product is not defined for any number of dimensions: it is in fact only defined for 3 and later for 7 dimensional spaces. This means that the above formula may be correct, but not for the general d-dimensional case. Probably who wrote this had the 3-dimensional formula in mind and thought that it could be extended to any number of dimensions.

Please correct me if I'm wrong, I'd be glad to learn about it. —Preceding unsigned comment added by Shirokuroneko (talk • contribs) 15:02, 24 October 2010 (UTC)


 * Note how We start by transposing the system to place C at the origin: this plane is two dimensional so we can choose some 3D subspace including the plane and calculate the cross products in that space. It does not matter which 3D subspace we choose as long as it includes the plane. It would probably be better to reformulate to remove the use of cross product which will fall out. --Salix (talk): 17:32, 24 October 2010 (UTC)
 * there is a problem with theese equation. because for the points [1,0,0][0,1,0][0,0,1] it gives wrong values of r and p0. For points [1,0,0] [0,1,0] [-1,0,0] result is correct 91.201.19.107 (talk) 17:24, 25 April 2023 (UTC)

splitting?
I was wondering should this article not be split in two seperate articles? one about Circumscribed circle (triangle) for all about the circumscribed circle of a triangle, and one about the general cirumscribed circle of polygons. Much of the article is only about triangles and other polygons are woefully under represented. WillemienH (talk) 09:56, 21 June 2015 (UTC)


 * I disagree with the proposal to split. Most of the content is about triangles' circumcircles because that's what we know most about. It would look strange to have an article called "Circumscribed circle" that didn't say much about the triangle case. And anyway, there are undoubtedly a substantial number of articles about triangles that link specifically to this one -- all those links would be wrong if the triangle material were to be split off. Loraof (talk) 14:11, 13 July 2015 (UTC)

Proof for circumcenter vector formula
The formula,


 * $$ U = \frac{a^2(b^2+c^2-a^2)A+b^2(c^2+a^2-b^2)B+c^2(a^2+b^2-c^2)C}{a^2(b^2+c^2-a^2)+b^2(c^2+a^2-b^2)+c^2(a^2+b^2-c^2)}.$$

stated in the "Article" page, can be proved as following.

Step1.
Let U be the circumcenter of ΔABC. If you draw lines from U to each of three vertices, three small triangles appear. As U is the circumcenter, lengths of now drawn lines should be equal. Let it be r. Each of three small triangles is isosceles, which also means related angles are equal. As the sum of inner angles of an arbitrary triangle is 180°,
 * ∠BCU = (180°-∠CUB)/2
 * ∠UCA = (180°-∠AUC)/2
 * γ = ∠BCA (definition)
 * = ∠BCU+∠UCA
 * = (360°-∠CUB-∠AUC)/2
 * = ∠BUA/2
 * → ∠BUA = 2γ.

Adding a perpendicular line from U to each line, each of isosceles triangles is divided into two mutually congruent (though reflective) rectangular triangles. Thus angles are assigned as Figure 1.

Step2.
Using r (distance between U and each vertex), and a, b, c (lengths of BC, CA, AB), areas of isosceles triangles SA, SB, SC are,
 * SA = (a r cosα)/2
 * SB = (b r cosβ)/2
 * SC = (c r cosγ)/2

Configuration here is expressed in Figure 2. Area of original triangle is,
 * S = SA + SB + SC  = r(a cosα + b cosβ + c cosγ)/2.

Step3.
Let's regard AB as a base line. As this is common to both the original triangle and ΔABU, proportion in their area is equal to proportion in their height. Adding a line that is parallel to AB and that passes U, inclined line CA (whose length is b) is divided into the same ratio as their height. So, over Figure 3,
 * m/b = SC/S = c cosγ/(a cosα + b cosβ + c cosγ).

In the same way, regarding CA as a base line,
 * n/c = SB/S = b cosβ/(a cosα + b cosβ + c cosγ).

Now using the parallelogram appeared at the below left corner, vector to U is got as the sum of three vectors,
 * U = A + (C-A)m/b + (B-A)n/c
 * = A + (C-A)c cosγ/(a cosα + b cosβ + c cosγ) + (B-A)b cosβ /(a cosα + b cosβ + c cosγ)
 * = (A a cosα + B b cosβ + C c cosγ)/(a cosα + b cosβ + c cosγ)

Using the Law of cosines,
 * U = { A a (b2 + c2 - a2)/2bc + B b (c2 + a2 - b2)/2ca + C c (a2 + b2 - c2)/2ab }/


 * { a (b2 + c2 - a2)/2bc + b (c2 + a2 - b2)/2ca + c (a2 + b2 - c2)/2ab) }
 * = { A a2 (b2 + c2 - a2) + B b2 (c2 + a2 - b2) + C c2 (a2 + b2 - c2) }/{ a2 (b2 + c2 - a2) + b2 (c2 + a2 - b2) + c2 (a2 + b2 - c2) }

Tsukitakemochi (talk) 14:09, 27 July 2015 (UTC) Tsukitakemochi (talk) 14:11, 27 July 2015 (UTC)

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Removed proof that every triangle is cyclic
The "Triangles" section previously contained this claimed proof of the fact that every triangle is cyclic:


 * This can be proven on the grounds that the general equation for a circle with center (a, b) and radius r in the Cartesian coordinate system is
 * $$(x - a)^2 + (y - b)^2 = r^2.$$


 * Since this equation has three parameters (a, b, r) only three points' coordinate pairs are required to determine the equation of a circle. Since a triangle is defined by its three vertices, and exactly three points are required to determine a circle, every triangle can be circumscribed.

This proof is clearly flawed, because three collinear points do not determine a circle but the proof does not require that the points be noncollinear.

In general, there is no reason that n points must determine a unique member of a family of curves described by an equation with n parameters.

I've removed this flawed proof from the article. —Bkell (talk) 01:12, 14 October 2017 (UTC)
 * I agree. The two-bisector construction provides a better proof. —David Eppstein (talk) 19:06, 16 October 2017 (UTC)

Horrible examples
There are too few examples in the article where some points are inside the circle rather than on it. Especially examples are bad where there are many points and none of them inside the circle. Such examples make readers think that you can always draw a circle that goes through all points. — Preceding unsigned comment added by 2003:EC:9731:FD00:E519:7499:4D69:1592 (talk) 09:18, 24 May 2021 (UTC)
 * You did see the part in the first sentence that the circumscribed circle "passes through all the vertices", right? If there are vertices inside, it is not a circumscribed circle and not an example. Maybe you are looking instead for smallest-circle problem? —David Eppstein (talk) 15:40, 24 May 2021 (UTC)

Alternative Construction diagram incorrect
I believe the alt construction diagram has the incorrect angles labeled 90-α. The construction should proceed by drawing the 90-α angle from the side adjacent to the angle α. That way it's projection will intersect the opposite side in a right angle and so run through the circumcenter. — Preceding unsigned comment added by Rpnman (talk • contribs) 18:51, 7 December 2021 (UTC)

Article should be split
This article's scope is too broad, and it ends up doing a poor job covering any of its included topics because the resulting organization is awkward. The circumcircle of a triangle, an essential topic linked to from all over is not given appropriate basic description in the lead section, even though it is what most readers are going to be looking for. The content under § Triangles dives right into a disorganized mishmash of specific details and does not do an adequate job of introducing or setting up the concept in a way that is legible and easy to follow for non-specialists.

In my opinion this article should be moved to circumcircle and should limit itself to describing the circumcircle of a triangle (and the perpendicular bisectors, circumcenter, etc.); everything in § Triangles can be moved to top level, which leaves more room for expansion, especially common proofs of some of the basics, a historical discussion, and so on. A section can be added about the circumcircle of triangles on the sphere and hyperbolic plane, a section can be added about the circumcircle in a pseudo-Euclidean plane of signature (1, 1), and so on.

The circumcircle of a general cyclic polygon should be discussed at a separate page cyclic polygon (we already have cyclic quadrilateral, which is an appropriate scope and title for an article).

The minimum bounding circle is already split to smallest-circle problem, again a fine scope for an article.

Thoughts? –jacobolus (t) 18:54, 15 December 2022 (UTC)
 * I agree that splitting off the content about triangles from the content about cyclic polygons makes sense. I also agree that circumcircle and cyclic polygon are good names for the split-off articles. Maybe rather than redirecting to any one of these, the current circumscribed circle title could become a disambiguation or set-index article pointing to both of those topics + smallest-circle problem? —David Eppstein (talk) 19:31, 15 December 2022 (UTC)
 * Having circumscribed circle as a disambiguation page sounds good to me. But if that's the result we decide on it should probably be accomplished by first moving this page to circumcircle, next copying relevant material over the redirect cyclic polygon (which was its own topic in 2003), and then finally creating a new article at circumscribed circle, to keep the edit history mostly intact. (Edit: or maybe the pages circumcircle and circumscribed circle could instead be swapped; both contain some relevant edit history.) –jacobolus (t) 19:53, 15 December 2022 (UTC)
 * I support this plan, although I don't think the page history of circumcircle is particularly valuable. Apocheir (talk) 22:57, 15 December 2022 (UTC)
 * Can someone with the appropriate technical privileges do this move? Splitting the text around shouldn’t be too tricky, but I think it takes some administrative privilege to move circumscribed circle &rarr; circumcircle when there is already a redirect there with some past history. –jacobolus (t) 18:32, 14 February 2023 (UTC)
 * Moves that need admin assistance typically go to Requested moves. Apocheir (talk) 23:27, 14 February 2023 (UTC)


 * While we are at it, perhaps we can move Incircle and excircles of a triangle -> Incircle and excircles for concision. –jacobolus (t) 05:51, 18 December 2022 (UTC)
 * I also did this one. –jacobolus (t) 17:45, 29 June 2023 (UTC)
 * Okay, @UtherSRG helpfully did the swap for us. I merged the material about cyclic polygons in general into concyclic points, as it seems to have substantially overlapping scope. Ping @David Eppstein, @Apocheir. Still working on making an index page out of circumscribed circle. –jacobolus (t) 17:41, 29 June 2023 (UTC)
 * David, can you help me properly format Circumscribed circle as an index/disambiguation page? –jacobolus (t) 18:15, 29 June 2023 (UTC)