# Tagged Questions

For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, angles.

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### Triangulation of clusters of points

I am trying to solve a triangulation problem, but I am not really sure what is the best way to tackle it. I have a series of points $P$ in an $n$-dimensional space. These points are clustered in $k$ ...
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### Sufficient condition for a simple polygon

Some background: I'm trying to simulate biological cells as polygons in 2D. Real biological cells have an internal cytoskeleton which enables them to conform to a variety of shapes, many non-convex. ...
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### Simple prove that product of the diagonals of a polygon = N

There is a beautiful fact: If you take a regular N-sided polygon, which is inscribed in the unit circle and find the product of all its diagonals (including two sides) carried out from one corner ...
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### If a line through the centroid $G$ of $\triangle ABC$ meets $AB$ in $M$ and $AC$ in $N$, then how to prove that $AN\cdot MB+AM\cdot NC=AM\cdot AN$ .

If a line through the centroid $G$ of $\triangle ABC$ meets $AB$ in $M$ and $AC$ in $N$, then how to prove that $AN\cdot MB+AM\cdot NC=AM\cdot AN$ .
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### Circle touching three tangential circles

The circles $C_1,C_2$ and $C_3$ with radii $1,2$ and $3$, respectively, touch each other externally. The centres of $C_1$ and $C_2$ lie on the $x$-axis, while $C_3$ touches them from the top. Find the ...
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### convex curve equivalent definition

I have seen two different version of definition of a closed convex curve in a plane: For a curve $r(t)=(x(t),y(t))$ 1.The whole curve lies on only one side of any tangent of the curve 2.Any ...
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### Proving that four points are non-concyclic

In the figure, G and H are the circum-center and the orthocenter of ⊿ABC respectively. AH produced meets BC at O. GR ┴ BC at R. BS is the diameter of the circumscribed circle. Show that B, O, H, and G ...
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### Find the maximum area of a parallelogram given that you know the perimeter

The perimeter of the parallelogram ABCD is 14, therefore 14=2(AB+AD) so AB+AD=7. I know that the sizes of AB, BC, CD and AD are natural numbers. How can I find the maximum area of the parallelogram? ...
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### Geometry Math Olympiad Question [closed]

In the diagram below, AD is perpendicular to AC and $∠BAD = ∠DAE = 12^\circ$. If $AB + AE = BC$, find $∠ABC$. The above is the diagram. I came across this question in a Math Olympiad ...
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### Area of ​​the intersection of two discs: Integral solution?

Here is the problem : We consider two circles that intersect in exactly two points. There $O_1$ the center of the first and $r_1$ its radius. There $O_2$ the center of the second and $r_2$ its ...
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### Hinge point in quadratic program (bilateral constraint)

My question itself is possibly quite simple and I guess that if someone can answer me they probably does not need a wall of text that is my background to the problem, but I figured I should provide as ...
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### If we inscribed all the 6 regular convex four-dimensional polytopes in a sphere, which one would have the highest volume?

When a dodecahedron is inscribed in a sphere, it occupies more of the sphere's volume (66.49%) than an icosahedron inscribed in the same sphere (60.54%). But what about for the 6 regular convex four-...
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Thank you. I just need them checked.
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### How to prove by induction the constructibility of a line segment of length $\sqrt{n}$?

How to prove the following statement by induction? If a line of unit length is given, then a line of length $\sqrt{n}$ can be constructed with straightedge and compass for each positive integer $n$. ...
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### What are the formulas for the number of vertices, edges, faces, cells, 4-faces, …, $n$-faces, of convex regular polytopes in $n \geq 5$ dimensions?

I know that in dimension $n \geq 5$ there are only 3 kind of convex regular polytopes in each dimension: the $n$-simplex, the $n$-cube and the $n$-orthoplex. I would like to know if there are ...
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### Weird inequality answer, truncate or round?

When arriving at the final answer for a double inequality question, it appears that my text book has truncated one part and rounded the other. Is there some weird inequality rule that I don't know ...
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### Why is a straight line the shortest distance between two points?

The first application I was shown of the calculus of variations was proving that the shortest distance between two points is a straight line. Define a functional measuring the length of a curve ...
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### Distances between two orthocenters

Let a Triangle $\triangle ABC$ be inscribed in a circle, along the arc $\overset{\frown}{BC}$ lies a point $P$ such as, $BP=4\sqrt{2}$. Compute the distance between the two orthocenters of the ...
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### Right-angled triangles on a graph

My question today is whether or not a concise formula has been discovered for the coordinates along the hypotenuse of a right-angled triangle when plotted on a graph. I have been working on this and ...
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### Finding the x-intercept of a straight line

My question today is whether or not the formula for the x-intersect has been discovered for any straight line on a graph. I have been working on this for a bit and I think I have discovered a formula ...
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### Why is it safe to approximate $2\pi r$ with regular polygons?

Considering this question: Is value of $\pi = 4$? I can intuitively see that when the number of sides of a regular polygon inscribed in a circle increases, its perimeter gets closer to the perimeter ...
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### How to find the equation of $X$?

A particle Q is moving at a constant speed $V$ in a circular path of radius $R$. P is a fixed point below $O$(the center) at a distance $r$. X$=$PQ How can I find an equation for $X$? It is actually ...
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### Juxtapose two triangles with a common edge

I'm not experto in geometry but I'm trying to do a software that handle triangles in various way. And I'm trying to learn geometry, of course : ) I have one fixed triangles $T1 = \hat{ABC}$ and a ...
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### How to check does polygon with given sides' length exist?

I have polygon with $n$ angles. Then I have got $n$ values, which mean this polygon's sides' length. I have to check does this polygon exist (means - could be drawn with given sides' length). Is ...
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### An attempt to Prove (or Disprove)

I just found out from my calculations the following: Corresponding a given length of a one-dimensional element, if a two-dimensional lamina, (having same boundary length as the length of the one-...
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### Calculate minimum width of lane

A lane runs perpendicular to a road $64 ft$ wide. If it is just possible to carry a pole $125 ft$ long from the road into the lane, keeping it horizontal, then what should be the minimum width of the ...
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### Perpendicular to Z axis or Skew to Z axis? (Definition of Perpendicular)

Question Part 1. Consider the following, where the point is the intersection of the sphere and a tangent plane. Consider a Euclidean coordinate system where: Blue dot is the origin (0,0,0). Z-...
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### Luzin's theorem, finding a continuous function under a certain condition

Let $X:\mathbb R^2\rightarrow\mathbb R,$ be the map defined by $(x,y)\mapsto y-x.$ Let $h:\mathbb R\rightarrow\mathbb R$ be Borel measurable. Let $\mu$ be a Borel probability measure on $\mathbb R^2.$...
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### Can loci be defined with more than 1 moving point?

So recently I've been playing around with some ideas in my head and wondering whether there are loci in which more than 1 point is movable and others are fixed. For example, I started with a circle ...
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### Reference request: algebraic methods in geometry

I am a (soon to be) third year undergraduate who has just finished courses in linear and abstract algebra. While I enjoyed the study of algebraic structures in their own right, my favorite part of the ...
"Let $\hat{ABC}$ be an isosceles triangle with $AB=AC$. $D$ is a point on $BC$ such that $DC=DB$ (middle of $BC$). $E$ is the projection of $D$ on $AC$ and $F$ the middle of $DE$. Prove, using vectors ...