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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Geometry problem with medians and areas

Let $ABC$ be a triangle with centroid $G$. A perpendicular line from $G$ to the line $BG$ intersects the parallel through $A$ to the line $BC$ in $D$. Prove that $AC\cdot BD\geq 2\cdot area[AGBD]$. So ...
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Create a 2D plane tessellation starting from given centroids

I have a problem: I want to find a method which, given a set of centroids as x,y coordinates (the number of the centroids does not matter, more than one) it returns ...
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Centroid of a parabolic arc

Find the centroid $C=(\bar{x},\bar{y})$ of the parabolic arc $y=16-x^2$ over $[-4,4]$. From symmetry, $$\bar{x}=0$$ To find $\bar{y}$, substitute $\tilde{y}=y$, $dL=\sqrt{1+4x^2} dx$ in $$\frac{\...
Starlight's user avatar
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Triangle area problem involving the centroid [closed]

If $X, Y, Z$ are the feet of the perpendiculars from the centroid to the sides $BC, CA, AB$, prove that the area of $\triangle XYZ=\dfrac{4\Delta^2(a^2+b^2+c^2)}{9a^2b^2c^2}$. Solve only using ...
Chinmay 's user avatar
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Locus of centroid of equilateral triangle inscribed in ellipse.

Problem Find the locus of the centroid of an equilateral triangle inscribed in the ellipse $x^2 / a^2 + y^2 / b^2 = 1$ My attempt I assumed 3 parametric points on ellipse P, Q and R. And assumed the ...
Mark's user avatar
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Unusual centroid of a figure

One day the question arose: how to determine centroid of annulus to be in the specified place? Of course, it can't be usual centroid as a mass center. We can use symmetry, radus and concentric circle ...
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Is the area on both sides of a centroid equal?

I am given the function $f(x)=x$ and I am trying to find the $x$ centroid between $0$ and $1$. I know the that $\bar x $ can be found by the formula $$\frac{\int_0^1xdA}{\int_0^1dA}$$ which then ...
Timmy Diehl's user avatar
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Getting a point in the interior of a polygon without relying on winding order?

I am given an arbitrary set of points embedded in 3D. The points are guaranteed to be ordered such that their order yields a simple closed polygon, but there is no information about whether they wind ...
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The midline of a triangle

Triangle $ABC$ is isosceles with $AB = AC$. $P$ is a variable point on $AB$, and $Q$ is a variable point on $AC$, so that $BP = AQ$. Let $O$ be the midpoint of $PQ$. Prove that $d(O, BC)$ is constant, ...
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Average of a function and center of mass

The geometric center of an object is the center of mass of an object with uniform density. The average height of a semicircular wire is $\pi R/4$ where R is the radius. But the actual y coordinate of ...
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Centroid of semi-circle using weighted avarage.

let the centroid be the point $(x_c,y_c)$ where $$x_c = \frac{\int x ds}{\int ds}$$ $$y_c = \frac{\int y ds}{\int ds}$$ Find the centroid of the semicircle $x^2 + y^2 = a^2$, where $y >= 0 $ I ...
SirMrpirateroberts's user avatar
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Different centroid values for trapezoid using two different methods?

Suppose I have the following trapezoid: Now, I used two methods to calculate the x coordinate of its centroid: $x_c = \frac{x_1 + x_2 + x_3 + x_4}{4} = \frac{0+0+3+4}{4} = 1.75$ The second method ...
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Geometric method of finding centroid of point cloud in plane

The cartesian coordinates of the centroid of a set of points in the plane is the mean of their cartesian coordinates. Is there a geometric way of finding the centroid of an arbitrarily large set of ...
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Centroid of a parabola with positive and negative areas [closed]

Consider the parabola $18x-3x^2-12$ (https://www.desmos.com/calculator/efel9y5dbj). It is required to find the centroid of this parabola contained between x=0 and x=2. For x=0 to x=0.764 its ordinates ...
L Manimaran's user avatar
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Mathematical Paradox: How Can The Center of a Shape Be Located OUTSIDE This Shape?

Recently I have been learning about Geospatial Analysis in which we are often interested in using computer software to analyze the mathematical properties and characteristics of polygons (e.g. ...
Uk rain troll's user avatar
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If BD + BC = 3BE, how can we prove G is the centroid of triangle ABC?

In the acute-angled triangle $\triangle ABC$, we mark the middle with $M$ side $BC$ and with $D$ the foot of the height in $A$. Let $G$ be a point on the median $AM$ and $E$ is its projection on the $...
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Visualise proof for relation between average (squared) distance to all points, and (squared) distance to centroid

Consider a set of real numbers $S=\{ a_1, a_2, \ldots, a_n\}$ and $\mu$ denotes the mean of all the points. Consider a number $z$, then it is known that - $$\sum_{i=1}^n (z-a_i)^2 = \sum_{i=1}^n (a_i -...
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What is the relation between a triangle’s centroid and its pedal triangle?

The pedal triangle of a point inside a triangle is the triangle formed by connecting the three feet of the perpendiculars drawn from that point to each side of the triangle. What is the relation ...
Alex Wang's user avatar
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Calculating Failure Angle of Silo Wedge

I am working on an optimization problem, specifically calculating the lateral earth pressures on the inside of a silo wall. I have been trying to solve for the failure angle of a silo wedge based on ...
MS_Engineer's user avatar
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Why does the centroid of a triangle stay in place when one side of it is rotated?

I was experimenting with GeoGebra and I found something that I don't quite understand. I wanted to see what would happen to the centroid of a triangle if one of its sides rotated inside a circle ($\...
Kourosh Fatehi's user avatar
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Circumscribed Circle And Inscribed Circle As Minima

Let $X\subseteq\mathbf{R}^2$ be open, bounded, non-empty and define $c_p$ as the set of points minimizing $\int_{x\in X} \|x-c\|^p$, so e.g. $c_2$ contains only the centroid of $X$. I have the ...
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Find the center of mass of an n-dimensional hemisphere (mass is uniformly distributed) where $x_n\ge 0$

I want to find the centroid of an n-dimensional hemisphere with a radius $a$. The hemisphere has uniformly distributed mass and I denote it as $B_+^n(a)=\{(x_1,\cdots,x_n):x_1^2+\cdots+x_n^2\le a^2 \ ...
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Find coordinates of centroid of an area

I would like to find the centroid $(x_c^A,y_c^A)$ of the orange area $A$, marked in the attached picture, i.e. of the area between $$ f(x)=\ln(x)\qquad\textrm{ and }\qquad g(x)=1, 0\leqslant x\...
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Geometry Question on the Circumcentre, Incentre and Centroid of a Triangle

Question If there is a triangle $ABC$ where its circumcentre, incentre and centroid are named $O, I, G$ respectively, $AB=c$, $BC=a$, $CA=b$, and the radii of the circumcircle and incircle of the ...
Cheese Cake's user avatar
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Proof about the properties of the centroid in a finite set?

The whole text was too long to fit in the title. I found this statement without much of a citation or proof: "The sum of the squared distances from every point to the centroid is equal to sum of ...
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The gravity center

Good morning, I want to prove that all points (A, B, C, D), as shown in the figure Image, compute the same center of gravity of the square. Is there a way to prove that by using the local coordinates ...
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The center of gravity of a hexagon [closed]

Good morning everyone, I am doing some researches and I wanted to have a proper answer concerning the centroid of a hexagon. Is it equal to the arithmetic mean of its vertex coordinates? And is it the ...
IsLearning's user avatar
1 vote
1 answer
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Integration Issue with Finding a Centroid

I've been brushing up on my understanding of centroids in 2-dimensions, and I chose to try to find $M_x$ of the region bounded by $$f(x) = \sin(x-\frac{\pi}{2})+3$$ and the x-axis, $g(x)=0$, from $x=\...
Zach Adams's user avatar
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1 answer
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Centre of mass with double integration. What is Moment $Mx$?

First post here. I'm having serious trouble understanding how the Moment $Mx$ is solved for in a typical Centre of Mass problem. So, many people online, are teaching methods to solve for $Mx$ that are ...
Insaan Abdulla's user avatar
2 votes
2 answers
136 views

Proving the "centroid" property and the existence of corresponding convex polyhedron in Minkowski Problem

Assume $P$ is a convex polyhedron embedded in $\mathbb{R}^{3}$, the faces are $\left\{F_{1}, F_{2}, \cdots, F_{k}\right\}$, the unit normal vector to the face $F_{i}$ is $\mathbf{n}_{i}$, the area of $...
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2 votes
1 answer
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Centroid coordinate for region

This is the region from which I want to calculate the centroid: $\{(x, y) \in \mathbb{R}^2 : 0 < 2x < y < 3-x^2\}$. I calculated the area for the region and it is $A = \frac{5}{3}$. Now, for ...
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Weighted center of mass by number of neighbors

As indicated by the tag, this is a reference request of a notion I thought of, but have not been able to find any literature on it. So, maybe someone here can link some resource. Suppose there are $n$ ...
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Vertices of a regular pentagon from center and side length?

I was wondering if it was possible to calculate the coordinates of every vertex in a regular pentagon using only its side length and its center? The pentagon's center will not be fixed at the origin, ...
Ballum's user avatar
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1 answer
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How to find distance between center of rectangle and bound with degree?

Given width and height of rectangle, coordinates of centroid and degree, how to find coordinates of a point extended from centroid with degree? In the below image, I want to calculate (??, ??). Any ...
Guk's user avatar
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2 answers
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How to calculate the centroid of a Polytope?

Given a polytope is divided into simplexes, is it correct to calculate the centroid of the polytope as the average sum of its simplex centroid coordinates
linker's user avatar
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3 votes
1 answer
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Prove that $AP \cdot AQ+CP \cdot CQ=BP\cdot BQ$

Let $G$ be the centroid of $\triangle ABC$. A line $M$ through $G$ intersects the circumcircle of $\triangle ABC$ at $P$ and $Q$, where $A$ and $C$ lie on same side of $M$. Prove that $AP \cdot AQ +CP ...
helloheyhi's user avatar
2 votes
1 answer
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Is this function monotonically increasing as $x_2$ increases?

Suppose I have a differentiable and continuous function $f(x)>0$, the monotonicity of $f(x)$ is unknown. Assume that $x_1< x_2< x_3 \in \mathcal{S}$, $\mathcal{S}$ is the domain of $f(x)$. ...
Tyke's user avatar
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How to construct the centroid of a quadrilateral

I know how to construct the centroid of a quadrilateral as mentioned here. But my question is different from that. We know that if points B,C,G are given in geometry plane and for locating point A ...
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Intersection of half angle with centroid

I am in calculus 2 and I recently came up with a question. With the formulas f(x)=n and g(x)=x^2 is there a value for n in which the half angle between the intersection of the parabola and the ...
Jeremy's user avatar
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Moment Question

I am looking for feedback on some points actually. I am looking at the centroid equation $\frac{\int x f(x) dx}{\int f(x) dx}$, assuming that the integrals are bounded, will not consider constants. I ...
Gufu SDMF's user avatar
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1 answer
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Find list of closest points around a point in 3D space.

I have a list of points (x1,y1,z1) ...(xn,yn,zn). I have another point in this space, (X,Y,Z). I want to find a subset of the original points that form a boundary around (X,Y,Z). Or in other words, a ...
Nishita's user avatar
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3 votes
3 answers
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In a triangle ABC, if certain areas are equal then P is its centroid

Let $P$ be a point in the interior of $\triangle ABC$. Extend $AP$, $BP$, and $CP$ to meet $BC$, $AC$, and $AB$ at $D$, $E$, and $F$, respectively. If $\triangle APF$, $\triangle BPD$, and $\triangle ...
user1991's user avatar
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1 answer
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Pappus centroid theorem and Hypercones.

The volume of a straight cone in $\mathbb R^3$ is usually find adding the circular sections orthogonal to the height. If the base has radius $R$ and the height is $h$ we have: $$ V_{C3}=\int_0^h \pi r^...
Emilio Novati's user avatar
2 votes
1 answer
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A system of particles distributed on the surface of a ball, what is the "center of mass" of them on the surface?

Suppose a system of $n$ particles distributed on the surface of a ball, what is the "center of mass" of them on the surface? Does the following optimization problem have an analytical ...
Mike Mathcook's user avatar
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1 answer
230 views

How does one derive the centroid formula for multiple shapes?

I know how to derive the formula for the centroid of n sets of finite points, each with $m_k$ points and centroid $C_k$. The formula is: $$C=\frac{\sum_{k=1}^{n}C_km_k}{\sum_{k=1}^{n}m_k}$$ However, ...
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k-means clustering identity

Suppose $x_1, x_2, \dots, x_n$ are distinct points in $\mathbb{R}^d$, $\mathcal{S} = \{ S_1, S_2, \dots, S_k \}$ is a partition of $\{ x_i : i \in [n] \}$ (with $k \leq n$), and $\mu_i = \frac{\sum_{x ...
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4 votes
1 answer
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Any interpretation for the fact that centroid = optimal point for maximising volume of such cuboid?

(Visualising a sample case via the image at the bottom) Consider a plane $\frac xa+\frac yb+\frac zc=1$ so that it intercepts with the axis at $(a,0,0)$, $(0,b,0)$ and $(0,0,c)$, $a,b,c >0$. Now ...
anrokhshan's user avatar
2 votes
1 answer
138 views

Thermal center of a set

The current Wikipedia article on the heat equation includes a gif which shows how the solution to the heat equation evolves over time if the initial data is the indicator function of a domain. ...
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Every $2$-colouring of the lattice points of $\mathbb R^m$ has $n$ monochromatic points whose centroid is a lattice point of the same colour

I was asked the question Prove that every $2$-colouring of the lattice points of $\mathbb R^m$ has a collection of $n$ monochromatic points whose centroid is a lattice point of the same colour Now, ...
Sayan Dutta's user avatar
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Lines from vertices through centroid of a triangle bisect opposite sides.

It is well known that the lines from the vertices through the centroid of a general triangle bisect the sides opposite to each vertex. Is there a simple geometrical proof for this? I've managed to ...
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