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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1answer
29 views

Petr-Douglas-Neumann theorem - centroids don't match

According to Petr-Douglas-Neumann theorem (Wikipedia article) after all n-2 steps (n-number of sides/vertices in the polygon) are complete, the result is a regular polygon whose centroid coincides ...
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1answer
20 views

How would one define the “center” of a closed, non-discrete 1d loop?

Centroids describe the average of all the discrete points on a curve (in 2d space), but why is it that the center of a circle or ellipse is what it is? Moreover, is it possible to determine the center ...
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1answer
19 views

Finding value of a when the centroid of a region (p,0) is bounded by $y^2 = 4px$ and x = a

The problem is basically the title, however I just got stuck when I did two solutions. Here is one of them: $$y^2 = 4px\\ f(x) = \sqrt{4px}\\ g(x) = -\sqrt{4px}\\ \bar{x} = p = \frac{M_y}{M}\\ M_y = \...
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1answer
31 views

Measuring symmetry / Is there a way to estimate the “centroid of an ideal shape” if I measure a skewed one?

How to measure asymmetry? Is there a way to estimate the "centroid of an ideal shape" if I measure a skewed one? I.e. I produce a measure of the centroid and it's "off", since the ...
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4answers
138 views

In a triangle prove that $IG \perp BC \iff b + c = 3 a$

Consider the triangle $ABC$. Denote the incenter with $I$, the centroid with $G$ and $AB = c$, $AC = b$, $BC = a$. Prove that $IG \perp BC \iff b + c = 3 a$. I have tried using centroid's and incenter'...
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1answer
45 views

Condition for a rigid body in equilibrium on a horizontal surface

The following question is from a past paper from Further Mathematics and it has been bothering me immensely I have spent hours cracking at a way to make sense of it, the question is in two parts ...
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1answer
24 views

If we transform a set of unit vectors in $R^n$ using some $n \times n$ permutation matrix, is the centroid also transformed by the permutation matrix?

Given a set of unit vectors $V$ in $R^n$, the set's centroid $c^V$, and some $n \times n$ permutation matrix $P$, we define $V'=\{Pv~|~\forall v \in V\}$. Is it true that $c^{V'} = Pc^V$? If so, can ...
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1answer
41 views

How to find angle at centroid of triangle by its edge lengths?

I need to write a program that takes length of a triangle's edges and calculates the angle $\angle APB$ ($P$ is the centroid of the triangle). Thanks for any help or clue.
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58 views

How would I find the limit of this recursive sequence?

The Question $$x_n(c)=\frac{\int_{1}^{c} x_{n-1}(b)y_{n-1}(b)db}{\int_{1}^{c} y_{n-1}(b)x_{n-1}(b)'db}, x_1(c)=\frac{\int_{1}^{c} xf(x)dx}{\int_{1}^{c} f(x)dx} $$ $$y_n(c)=\frac{\int_{1}^{c} y_{n-1}(...
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22 views

Is it possible to express an infinitesimal area as a sum of squares?

I came across a question which was asking the center of mass of an area, the centroid. The proposed solution uses weighted average, the formula is that given a planar region, the x and y coordinates ...
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3answers
41 views

Area of a triangle formed by mid-points of two sides and the other vertex

Let the medians $AK$ and $BM$ of a triangle $ABC$ intersect at $O$, $AB=13,BC=14,CA=15$. Find the area of $\triangle AOM$. The lengths of the medians are given by:$$m_a=AK=\dfrac12\sqrt{2b^2+2c^2-a^2}...
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4answers
97 views

Show for the centroid $M$ of a triangle $ABC$ that $\vec{OM}=\frac13(\vec{OA}+\vec{OB}+\vec{OC})$

Show for the centroid $M$ of a triangle $ABC$ that $$\vec{OM}=\dfrac13\left(\vec{OA}+\vec{OB}+\vec{OC}\right)$$ where $O$ is an arbitrary point. I haven't studied position vectors and am very new to ...
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1answer
30 views

Twisted ribbons and centre of gravity

NOTE After posting this question, I realised I had not defined the overall shape of the form. The final note at the bottom was intended to correct that omission. I am no longer certain how this ...
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1answer
33 views

Finding the centre of mass of two right cones joined combined

The following question is given in my textbook Two uniform cones with base radius $r$ are joined together by their plane faces.Their lines of symmetry are aligned, the height of one cone is $6r$ and ...
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1answer
39 views

Centroid of the Region bounded by the functions: $y = x, x = \frac{64}{y^2}$, and $y = 8$.

Can someone help and teach me how to solve this problem? Find the centroid of the region bounded by the graphs of $$\begin{align} &y = x\\ &x = \frac{64}{y^2}\\ &y = 8 \end{align}$$ ...
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2answers
27 views

Finding the center of mass of a composite lamina

The following lamina is given and I have to find the distance from center of mass from AB and AC I started by splitting the lamina into two shapes, one rectangle of width $4a$ height $5a$ and one of ...
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0answers
62 views

Centroid of semi-circular arc vs Average of coordinates

Equation for the centroid a semi circular arc is x=2*r/(pi). If I have a circle with 1 cm of radius then its centroid is 2r/(pi)= 0.637 cm. Now I use the equation of this circle x^2+y^2=1^2 in Matlab ...
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2answers
68 views

Centroid and “Centre of mass”

I thought that the centroid was also the point at which the area in all directions was net 0. Aka area above the cenrroid was the same area as below. But for a triangle everything I can see seems to ...
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0answers
29 views

Given x, y coordinates of a path, find the point that minimizes the interquartile range of radii away from that point.

I have a list of x, y ∈ [-100, 100] coordinates that form a path. Example path I wish to find the point x,y ∈ [-100, 100] that minimizes the interquartile range (IQR) of radii away from the point. I ...
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2answers
31 views

Computing the centroid of some $n$ points in 3-dimensions

Solving for the centroid of some set of points in 2-dimensions is trivial. I'd like, however, to determine the centroid in 3-dimensions, given an arbitrary set of $n$ points. How can this most easily ...
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1answer
44 views

Can irregular polygons with unique vertices be guaranteed of a unique center point? Formula?

To visualize this, I've created the following diagram. I have sets of 80 numbers each. Here, just to draw it easily, I'm showing them as sets of 12 numbers each. The sets ...
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0answers
57 views

Calculating the centroid of a tetrahedron

Hope doing well and being healthy. I have a basic question on centroid of tetrahedrons. Are the coordinate of the centroid always the averages of $x$ and $y$ and $z$, by which I mean $$\frac14(x_1+...
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1answer
218 views

Find the center of mass of a quarter circle given by $\sqrt{r^2−x^2}$, $x\in[0,r]$

Q: The center of mass of the quarter circle given by $y=\sqrt{r^2−x^2}$, $x\in[0,r]$ is the point $(x, y)$. My first thought was that the circle was bounded by the points $(0,0)$, $(0,r)$, $(r,0)$ and ...
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1answer
51 views

Characterization of the center of a polygon

Let $O$ be the circumcenter of the regular polygon $P_n$. Then for any $A\in P_n$ one has $d(A,O)\leq r$, where $d$ is the usual Euclidean distance and $r$ is the polygon's circumradius. Prove that ...
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1answer
23 views

Computing a 3D quadrilateral inscribed sphere at its center?

For my game, I need to know when the player is over a powerup and for this I use a sphere collider: 3rd side view: Legend: orange, the powerup outline, a quad for which I know its centroid green, ...
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1answer
225 views

The centroid of the zeros of the kth partial sum of exp(z) is -1?

The question is the next: Let $P_k=1+z+\frac{z^{2}}{2!}+...+\frac{z^{k}}{k!}$, the kth partial sum of $e^{z}$. (a) Show that, for all values of $k\geq 1$, the centroid of the zeros of $P_{k}$ is -1. (...
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1answer
23 views

Does a plane passing througth the centroid of two points in euclidean 3D space must contain the two points?

Assume we have two points A and B in euclidean 3D space. The centroid of these two points is C. Now assume we define a plane P by means of the centroid C and an arbitrary normal vector n. Does the ...
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2answers
70 views

Solve for the base of an isosceles triangle

The area of an isosceles triangle is $S$ and the angle between the medians to the legs, facing the base, is $\alpha$. Find the base of the triangle. Let $CH$ be the third median of the triangle. The ...
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2answers
36 views

Formula for centroid of triangular number figurate

Triangular numbers are of the form T1 = 1 T2 = 1+2 T3 = 1+2+3 these can be represented as figurates. for example, a pool game has 15 balls = 1+2+3+4+5 = T5 ...
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0answers
57 views

Find centroid of a curve rotated at an angle.

I am trying to find the centroid of a convex curve rotated at an angle by $\frac{3}{8}$ Simpson's rule (I have curve points, not the curve equation). The curve is intersected by a line $y=c$ (centroid ...
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1answer
105 views

Let G be the centroid of a triangle $ABC$, $P$ any point in the plane, prove that $|AP|^2 +|BP|^2 + |CP|^2=|AG|^2+|BG|^2+|CG|^2+3|PG|^2$

Let G be the centroid of a triangle $\Delta ABC$ and $P$ any other point in the plane, prove that $|AP|^2 +|BP|^2 + |CP|^2=|AG|^2+|BG|^2+|CG|^2+3|PG|^2$ Let the coordinates be: $A=(x_1,y_1),B=(x_2,y_2)...
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2answers
77 views

Centroid of volume of revolution

Consider a solid generated by the curve $y^2 =ax^2+2bx+c$,rotated about the $x$-axis, and two plane surfaces at right angles to the latter, distance $h$ apart, and with areas $A$ and $B$. To prove ...
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1answer
34 views

Centroid of a region R using Average Values and Double Integrals

For a given function $f[x,y]$ over a region $R$ consisting of the points with $-3 ≤ x ≤ 3$ and $-2 ≤ y ≤ 2$ to calculate the centroid of a region we do the following: The centroid of a region $R$ in ...
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2answers
46 views

How do I solve this geometry problem related to the baricenter of a triangle?

The problem states: $ABCD$ is a square with side $a$. $E$ and $F$ are considered midpoints of sides $AD$ and $AB$ respectively. Let $G$ be the point of intersection of the $EC$ and $DF$ segments. Show ...
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2answers
68 views

Centroid of a quadrilateral $ABCD$: $\overrightarrow{GA} + \overrightarrow{GB} + \overrightarrow{GC} + \overrightarrow{GD} = \overrightarrow{0}$

Let $ABCD$ be a quadrilateral and let $G_1$, $G_2$, $G_3$, $G_4$ be he centroids of triangles $ABC$, $BCD$, $CDA$ and $DAB$, respectively. Assume without proof that $G_1G_3$ and $G_2G_4$ always ...
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0answers
19 views

Centroid of polytope with less vertices thand the dimension

I have polytope defined by $m$ vertices in $\mathbb{R}^{n}$. I would like to calculate their centroid (in the field of application this would be interpreted as an expected value assuming uniform ...
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2answers
83 views

Geometrical interpretation of $\frac{\sin(\alpha)}{\alpha}$

What is the geometrical interpretation of this quantity $$\frac{\sin(\alpha)}{\alpha}$$ such that $\alpha\in(0,\pi/2)$ For example, this is the abscissa of the centroid of the arcs of the unit circle ...
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1answer
116 views

Given the vertices of a convex polytope, calculate its centroid

I would like to calculate the centroid (center of mass in case of homogeneous materials) of a convex polytope (equivalent of polyhedron in $n-$dimensional space). The vertices are given and I can use ...
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1answer
272 views

Significance of Centroid of a Gaussian distribution

I am learning some of the parameters used for a gaussian function (or normal distribution). What is a centroid really? Is that just referring to the point where the y is the highest? What are some ...
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0answers
32 views

Bisectors and Incenter of a triangle

Let $G$ be the barycenter, $I$ is the incenter, $R$ the circumradius, $r$ the inradius and $p$ the semiperimeter of a triangle. Prove that $$GI^2=\frac{p^2+5r^2-16Rr}{9}.$$ I did this question using ...
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3answers
60 views

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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1answer
34 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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1answer
199 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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0answers
42 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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1answer
51 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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59 views

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
112 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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0answers
21 views

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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1answer
130 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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0answers
36 views

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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