# 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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### 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 ...
• 13
1 vote
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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 ...
• 13
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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 ...
• 181
1 vote
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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 ...
• 131
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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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### 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 ...
• 187
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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 ...
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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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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 $... 3 votes 1 answer 160 views ### What is the relation between a triangle’s centroid and its pedal triangle? The pedal triangle of a point inside a triangle is the triangle formed by connecting the three feet of the perpendiculars drawn from that point to each side of the triangle. What is the relation ... 1 vote 1 answer 91 views ### Calculating Failure Angle of Silo Wedge I am working on an optimization problem, specifically calculating the lateral earth pressures on the inside of a silo wall. I have been trying to solve for the failure angle of a silo wedge based on ... 0 votes 0 answers 54 views ### Why does the centroid of a triangle stay in place when one side of it is rotated? I was experimenting with GeoGebra and I found something that I don't quite understand. I wanted to see what would happen to the centroid of a triangle if one of its sides rotated inside a circle ($\...
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 ...