Questions tagged [centroid]

"The centroid or geometric center of a plane or solid figure is the arithmetic mean ("average") position of all the points in the shape. " This tag is for questions about the centroid of a geometrical shape, its properties and computation.

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In a triangle, G is the centroid of triangle ADC. AE is perpendicular to FC. BD = DC and AC = 12. Find AB.

G is the centroid of the triangle ADC. AE is perpendicular to FC. BD = DC and AC = 12. Find AB. According to the solution manual, we can let the midpoint of AC be H. D, G, and H are collinear as G is ...
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Coordinates of the centroid of volume

Determine the coordinates $$(x,y,z)$$ of the centroid of volume of this composite machine element. Here is the machine element: machine I know you have to use some integral with dV but I'm not sure ...
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1answer
17 views

Centroid coordinates of an irregular quadrilateral within a rectangular plane

I am developing a robotic project; the robot moves within a rectangular area 80cm x 180cm on a level horizontal surface. The area is bounded by four vertical walls, the robot has onboard four ultra-...
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12 views

How to determine that a cluster of points is actually two separate clusters?

I have a cluster of points defined by $x$ and $y$ coordinates. I know I can work out the center of the cluster by taking the average of $x$ and the average of $y$. Sometimes when I plot those points ...
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1answer
61 views

In triangle $ABC$, $G$ is the centroid, $I$ is the incenter, $GI$ || $BC$, what is $\frac{AB+AC}{BC}$?

In triangle $ABC$, $G$ is the centroid, $I$ is the incenter, $GI$ || $BC$, what is $\frac{AB+AC}{BC}$? I have little to no idea what to do with this problem. I drew the diagram and called the point ...
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35 views

Coordinates of centroid of a triangle

If $A(x_1,y_1) , B(x_2,y_2) , C(x_3,y_3)$ be the vertices of a triangle then prove that coordinates of centroid are given by $(\frac{(x_1 + x_2 + x_3)}3 , \frac{(y_1 + y_2 + y_3)}3)$
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45 views

Centroid Math Notes

Was there a typo in the notes? I noticed that when calculating the center of mass for y they do not square twice as shown in the formula. Can anyone confirm this?
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find centroid of hyperpyramid

I'm having trouble computing centroid of hyperpyramid (assume we have n points in n dimension). I already search a lot and I find how can calculate triangle and pyramid centroid, but I don't know how ...
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1answer
100 views

Triangle inscribed in a circle,2 points fixed and 1 moving. The track of centroid makes a circle but how do I prove it without cartesian coordinate?

Triangle ABC and circle O. A and B are fixed, but C is moving on the circle. So I have triangle ABC and circle O. A and B are fixed on the circle, but C is moving around the circle. Let G is the ...
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What is the centroid of the area bounded by x^3 and 10-x?

I understand what the problem is asking, but I don't fully understand the concept behind it. I know the formula for x bar is 1/A times the integral of x(f(x)-g(x)), and that the formula for y bar is ...
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64 views

Constructing equilateral triangle from three equilateral triangles.

I have heard about this problem some time ago and don't remember all the details.Therefore problem's conditions may have some mistakes. Problem: We have three arbitrary (size of edges, location and ...
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Finding the area of a triangle if the distances from the centroid to the vertices are $3$, $4$, $5$ [duplicate]

Let $G$ be the centroid of $\triangle ABC$, and let $AG=3$, $BG=4$, $CG=5$. Find the area of $\triangle ABC$. So I calculated the lengths of the diagonals which is $9$, $12$, $15$, but now I don't ...
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66 views

Get vertex value that shapes 90 degrees with triangle centroid

I have this triangle: I'm trying to get the vertex value highlighted in the green circle in order to draw that red line. Is there any equation that I can use to extract that value? ...
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1answer
56 views

Centroid of area enclosed by $x^n$ and $x^{1/n}$.

I am interested in finding the centroid of the surface enclosed by the lines $y=x^n$ and $y=x^{1/n}$ where $n \in \mathbb N, n >1$. The exercise, gives the following sketch. I know that in order ...
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Why is k proportional to centroid linkage distance mean and variance in k-means?

I've noticed that if I'm doing k-means clustering (in MATLAB) on any set of data, the mean and variance in centroid linkage distance appears to always be proportional to k. Is k always proportional ...
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12 views

Is the number of convex regions proportional to the inter-region centroid variance?

Say I completely divide some euclidean space into n convex regions for different values of n. Veronoi cells would be a good example. Say I take a subset of m regions. It seems almost obvious to me ...
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54 views

Locus of centroid of a $\triangle{ABC}$ with $A=(\cos\alpha,\sin\alpha)$, $B=(\sin\alpha,-\cos\alpha)$, and $C=(1,2)$

If $A(\cos\alpha, \sin\alpha)$, $B(\sin\alpha, -\cos\alpha)$ and $C(1,2)$ are the vertices of $\triangle ABC$, find the locus of the triangle's centroid as $\alpha$ varies. Let centroid be $(h,k)$, $$...
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29 views

Centroid of a plane figure of a plane figure.

A plane figure is enclosed by the parabola $y^2 =4x$ and the line $y=2x$. Determine the position of the centroid of the figure. Here is what I tried : Plotted the graph. $y=4ax, y=2x \\ 4x(x-1)=0 \...
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100 views

Finding centroid's coordinates using Pappus theorem

The task is to find the centroid of the given triangle (see the image above). We also should use the fact that the volume of a cone of radius $r$ and height $h$ is $V = \frac{1}{3}\Pi r^2h$. My ...
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197 views

How to use Vectors to find the centroid of a tetrahedron? [closed]

Suppose that four points - A,B,C,D (with position vectors a,b,c,d) are the vertices of a tetrahedron. And the mid points of BC, CA, AB, AD, BD, CD are denoted by P,Q,R,U,V,W. Using these info, I ...
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150 views

The centroid of a set lies in its convex hull

I've come across the following claim on a Wikipedia page about the centroid: The geometric centroid of a convex object always lies in the object How should I prove this claim for convex subsets of ...
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1answer
26 views

Area moments of a Freeman chain contour

The area moments of a polygon can be computed by generalizations of the shoelace formula for area. In particular, the first and second order moments are given by https://en.wikipedia.org/wiki/...
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135 views

Finding Centroid of a curve

$$x=e^t , y = \sqrt{2} t , z= \ e^{-t} ; ~~ ~~ 0\leq t \leq 1$$ Find the centroid of this curve. I know one formula for centroid which is $x = \frac{\int {x dx dy}}{\int dx dy}$ and for y as well, ...
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45 views

Two parallelograms and a centroid

A parallelogram $ABCD$ is given. The points $M_1$ and $M_2$ are on $AC$ and $M_3$ and $M_4$ are on $BD$ such that $AM_1 = CM_2 = \frac{1}{3}AC$ and $BM_3=\frac{1}{3}BD$. If $M_1M_3M_2M_4$ is a ...
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2answers
59 views

An identity associated with the centroid of a triangle

Four year ago, I am looking for a proof of my identity as follows: Let $ABC$ be a triangle, let $G$ be the centroid of $ABC$. Let $P$ be any point on the plane. Let $H, N, O$ on the plane such that: $...
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123 views

How do I find the moments and center of mass of a Lamina with given density?

I am trying to solve for the center of mass of the shape given below. I started by finding the area of the shape, which is $A = Atriangle + Acircle/4$ $ = 1/2 + pi/4$ $=(2+pi)/4$ then I used the ...
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4answers
104 views

Calculate coordinates of a centroid of $ABC$

We have a triangle $ABC$, where on the cartesian coordinate system: $A$ lies on $[-3, -2]$, $B$ lies on $[1, 1]$, $C$ lies on $[0, -6]$. How do we calculate coordinates for the centroid of this ...
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38 views

Proof that the triangle formed by mass centers is equilateral.

Let $ABC$ be a triangle and consider $A_1$, $B_1$, $C_1$ outside the triangle such that triangles $ABC_1$, $BCA_1$ and $ACB_1$ are equilaterals. Consider now $A_2$, $B_2$ and $C_2$ mass centres of $...
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149 views

Area of the intersection of two triangles.

Let $\triangle{ABC}$ be a triangle with $AB=5$, $BC=7$, and $CA=4$. Define $D$, $E$, and $F$, to be the midpoints of $AB$, $BC$, and $CA$ respectively. Let $G$ the intersection of the medians of $\...
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1answer
95 views

Intuitive Explanation to Pappus Theorem

Pappus's theorem is as follows: First theorem: The first theorem states that the surface area A of a surface of revolution generated by rotating a plane curve C about an axis external to C and on ...
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428 views

Centroid within non-convex 2d polygon

The centroid of an object is defined as the arithmetic mean of all points of the object. For non-convex objects, the centroid is often not a part of the object itself: Is there a definition of a ...
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1answer
51 views

centroid algorithm robust to missing poins

I need to find a center point of a person given the coordinates of all the joints. The joints of a person can be represented as a nodes of a graph with a fixed structure. The catch is some of the ...
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2answers
79 views

$z_1^2+z_2^2+z_3^2=3z_0^2$ if $z_1,z_2,z_3$ be the vertices of an equilateral triangle and $z_0$ be the circumcentre

Prove that, if $z_1,z_2,z_3$ be the vertices of an equilateral triangle and $z_0$ be the circumcentre, then $z_1^2+z_2^2+z_3^2=3z_0^2$ My Attempt $$ z_0=\frac{z_1+z_2+z_3}{3}\implies 3z_0=z_1+z_2+z_3\...
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51 views

How to solve this particular task about centroids

Can anyone help me with 29.I know how to find centroids when one function is given but in this one I don't think that knowing only the function of circle or rectangle will help us.I found that there ...
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1answer
56 views

How to find the best fit point inside a cluster?

I have a cluster with many points. Like this: Where I can visually identify a cluster of points and a ...
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63 views

Find the total mass and COM of a body

So in this exercise I am asked to find the center of mass of a certain solid defined by 2 elipsoids that have the following parametrizations: Elipsoid A: $$x=\cos\alpha \sin\theta\\ y=\frac{1}{2}\...
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206 views

How to find the barycenter/center of mass of this 2d shape without calculation?

I am given this shape (or any similar shape which is symmetrical). How can I determine the barycenter without calculating anything? My thought intuitively is, that the center should be at C, since it ...
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1answer
407 views

Coordinates of the centroid in a 3D rectangle

I have a 3D rectangle and I have to find the 3D coordinates of its centroid. I tried to take 4 vertices, one let's say the origin $o$ and the three adjacent vertices $a, b, c$. I computed the ...
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1answer
50 views

Finding lengths of sides on triangles with 3 given medians and lengths

The medians of $△TUV$ are $\overline{TX}, \overline{UY},$ and $\overline{VW}$. They meet at a single point $Z$. In other words, $Z$ is the centroid of $△TUV$. Suppose $\overline{UY}=33$, $\overline{TZ}...
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687 views

Finding the centroid of a tetrahedron

I have four points to form a tetahedron $$A=(0,-\frac{1}{2},-\frac{1}{4}\sqrt{\frac{1}{2}}) \\B=(0,\frac{1}{2},-\frac{1}{4}\sqrt{\frac{1}{2}}) \\C=(-\frac{1}{2},0,\frac{1}{4}\sqrt{\frac{1}{2}}) \\D=(\...
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123 views

Volumetric center of a polygon?

Is there a quick and efficient algorithm for calculating the volumetric center (probably the wrong term) of a polygon like that shown in the figure below somewhere around the blue dot? I'm not ...
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128 views

Can a spiral have its centroid at the origin?

A spiral is a curve $\gamma$ with the polar equation $r=f(\theta)$ where $f$ is a continuous positive strictly monotone function on some interval $[a, b]$, $-\infty<a<b<\infty$. Best known ...
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1answer
58 views

K-Means equality proof

Is there any geometrical and or short proof for the equality ${\displaystyle {\underset { }{ }}\sum _{i=1}^{k}\sum _{\mathbf {x} \in S_{i}}\left\|\mathbf {x} -{\boldsymbol {\mu}}_{i}\right\|^{2}}={\...
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How to find optimum (least square error) “square splitting circular arc” with given two centroids?

I am looking for an analytic solution for the following problem. A unit square is given. Coordinates of two points inside the square are given. What is the best circular arc which splits the square in ...
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151 views

Convex sum order

If I have a strictly convex function $f(x)$ with $f''(x)>0$ and if I know that for some $a\le b \le c$ and $x \le y \le z$ I have $$a+b+c = x+y+z$$ $$f(a)+f(b)+f(c)=f(x)+f(y)+f(z)$$ can I ...
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108 views

Centroid of a non-polygonal concave shape determined by an equilateral triangle an its incircle

I have a diagram here of an equilateral triangle ABC, centre O, where circle centre O has tangents which are all three sides of the triangle ABC. M is the midpoint of AB, and F is the intersection of ...
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259 views

Better “centerpoint” than centroid for placing a map marker inside a concave region (that may have holes)?

I'm using the centroid of polygons to attach a marker in a map application. This works definitely fine for convex polygons and quite good for many concave polygons. However, some polygons (banana, ...
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92 views

Line through triangle's vertex-centroid midpoint

I've been struggling for a couple of days trying to solve this geometry problem. Here it is: On this triangle, $G$ is its centroid (meaning that $AM = MC$). Also, $BN = NG$. He wants to know the ...
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407 views

What is the locus of centroid of triangle OAB?

A variable circle having fixed radius $a$, passes through origin and meets the coordinate axes in point $A$ and $B$. What is the locus of centroid of triangle $OAB$, $O$ being the origin?
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570 views

Centroid of quadrilateral on coordinate plane

I'm having trouble understanding the motivation between finding the centroid of a quadrilateral. Q: Find the centroid of a quadrilateral with vertices at (-8,12), (7,15) (13,-9), and (-2,-3). I've ...