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.
301 questions
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Finding the angles of a non-equilateral $\triangle ABC$ with centroid $G$ such that $\angle GAB=\angle GCA=30^\circ$
In the attached figures, $G$ is the centroid of $\triangle ABC$.
When this triangle is equilateral ( fig 1) , it is obvious that each of the two angles shown in this figure measures $30^\circ$.
Using ...
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How can we calculate the radius of the circumcircle of triangle ABC under these conditions?
We know that if H is the orthocenter of triangle ABC , then the circumcircles of triangles HAB , HAC and HBC all have the same radius R : the radius of the circumcircle of ABC .
So , I asked myself ...
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A rectangular circumhyperbola through the orthocenter has its center collinear with the centroid and antipode of fourth intersection point
The problem is related to the post Prove: A triangle inscribed in a rectangular hyperbola has its orthocenter on that hyperbola, the post Prove that a conic section through the vertices of a triangle ...
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Finding the centroid of an arc of a curve defined in polar coordinates [closed]
Find the position of the centroid of the arc of the semi cardioid ( $r = a(1 + \cos \theta) )$
Options are
(A) $( \left(\frac{a}{5}, \frac{a}{5}\right) )$ )
(B) $( \left(\frac{2a}{5}, \frac{2a}{5}\...
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Labeling of classical triangle centers: Why $H$ for orthocenter, $G$ for centroid, $O$ for circumcenter?
I was learning about triangles, and I came to know that orthocenter is represented by $H$, centroid is represented by $G$, and circumcenter is represented by $O$.
I don't understand why these are ...
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The length between orthocenter and intersection point of two circles
Here the construction of the diagram.
Draw a circle $C_1$ as incircle of triangle $ABC$. $I$ is the incenter and H is orthocenter of triangle $ABC$.
Draw circle $C_2$ as circumcircle of triangle $BIC$...
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Centroid of a set of vectors comprising polyhedral vertices
Consider the constant-norm vectors corresponding to the vertices of a regular polygon. The centroid of a set of these vectors, including duplicates (for certain $n$-gons belonging to the larger ...
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Barycenter of a triangle using the shoelace formula
Consider a triangle with vertices $\{V_i=[\xi_i\,\,\eta_i]'\}_{i=1}^3$. As is well known, the barycenter (or centroid) $g$ of a triangle coincides with the mean of its vertices, so that \begin{...
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How can we find the position vector of the centroid G in terms of a, b, and c?
The problem
Consider a triangle ABC. Find, with proof, the position vector of its centroid G, with
respect to the position vectors of A, B and C. Suppose the side lengths of the triangle are
a =BC, b=...
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Prove that $M$ is the centroid of the triangle $BCD$.
the question
Consider the tetrahedron $ABCD$ and $M$ a point inside the triangle $BCD$. Parallels taken from $M$
to the edges $AB$, $AC$, $AD$ intersect the faces $(ACD)$, $(ABD)$, respectively, $(ABC)...
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On the shoelace formulas
I've stumbled on this nice formula to compute the barycenter $\bar{c}$ of an arbitrary (but not self-intersecting) polygon
\begin{equation}
\bar{c} \triangleq \sum_{i=1}^n \frac{A_i}{A} \bar{c}_i
\...
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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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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
$$\frac{\...
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Locus of centroid of equilateral triangle inscribed in ellipse.
Problem
Find the locus of the centroid of an equilateral triangle inscribed in the ellipse $x^2 / a^2 + y^2 / b^2 = 1$
My attempt
I assumed 3 parametric points on ellipse P, Q and R. And assumed the ...
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Is the area on both sides of a centroid equal?
I am given the function $f(x)=x$ and I am trying to find the $x$ centroid between $0$ and $1$. I know the that $\bar x $ can be found by the formula $$\frac{\int_0^1xdA}{\int_0^1dA}$$ which then ...
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The midline of a triangle
Triangle $ABC$ is isosceles with $AB = AC$. $P$ is a variable point on $AB$, and $Q$ is a variable point on $AC$, so that $BP = AQ$. Let $O$ be the midpoint of $PQ$. Prove that $d(O, BC)$ is constant, ...
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Centroid of semi-circle using weighted avarage.
let the centroid be the point $(x_c,y_c)$
where $$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 ...
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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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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. ...
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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 ...
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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 ($\...
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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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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 ...
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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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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=\...
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