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

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106 views

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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0answers
59 views

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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4answers
1k views

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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1answer
72 views

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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2answers
444 views

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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2answers
272 views

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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2answers
95 views

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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1answer
771 views

Analytic geometry - rotation + translation

In $K=O\vec{e_1}\vec{e_2}\vec{e_3}$ I have to find the analytical representation of the screw motion( rotation + translation) $\psi$ with a rotation axis $g$ given by the points $A(5,-4,3)$ and $B(0,0,...
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1answer
325 views

Find the equation of a plane that is perpendicular to another plane, parallel to a line and goes through a point

Find the equation of a plane which is perpendicular to the plane $$\pi_1\equiv x-3y-z+1=0,$$ parallel to a line $$l\equiv\frac{x - 2}{2} = \frac{y -3}{-3} = \frac{z}{1}$$ and goes through point $P = (-...
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2answers
1k views

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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1answer
186 views

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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2answers
337 views

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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0answers
47 views

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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1answer
88 views

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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1answer
111 views

Could I please have my answers checked?

Thank you. I just need them checked.
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1answer
160 views

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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1answer
232 views

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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0answers
48 views

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 ...
23
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3answers
11k views

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 ...
3
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2answers
67 views

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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1answer
494 views

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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1answer
43 views

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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4answers
87 views

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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3answers
54 views

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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1answer
140 views

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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2answers
322 views

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 ...
0
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1answer
42 views

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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1answer
113 views

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 ...
1
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1answer
38 views

Prove that $f(A)\leq max(f(P),f(Q),f(R))$

Consider any $\bigtriangleup PQR$ in the $x-y$ plane. Let $f(x,y)=ax+by+c$ , where $a,b,c\in\mathbb{R}$. Let $A\in\mathbb{R^2}$ be any point in the interior or on the $\bigtriangleup PQR$. Prove that $...
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2answers
78 views

Find $\angle BOD$ in the given figure.

Consider a circle with centre $O$. Two chords $AB$ and $CD$ extended intersect at a point $P$ outside the circle. If $\angle AOC=43^\circ$ and $\angle BPD=18^\circ$, then the value of $\angle BOD$ is ...
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3answers
108 views

$ \frac{dy}{dx} = \tan(a) $ ; the derivative of a circle at a point (tangent) with respect to $y$ and $x$

warm up derivative: slope of the tangent line for the top half of a circle. I found that $$ \dfrac{dy}{dx} \left( x^2 + y^2 = 1 \right) \\ \rightarrow \dfrac{dy}{dx} = -\dfrac{x}{y} $$ please excuse ...
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1answer
19 views

Eigenvector shared by two endomorhisms

I am guessing if the following fact is true: Let be $V$ a finite vector space above a field $K$. Let $f, g$ be two endomorphisms of $V$ with $f g = g f$. We assume that both $f$ and $g$ have got at ...
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2answers
71 views

Don't know what to do [closed]

In the family R of all triangle on the same base whose areas are all equal prove that the isosceles triangle in R has the least perimeter
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1answer
39 views

What is the Z thickness of a 2D (X,Y) plane that is instantly transported into a 3D space?

Assuming I had a two-dimensional clipped plane having size $(1.0, 1.0)$ along $(X,Y)$; what would the $Z$ value become if this plane were transported, instantly, into a three-dimensional world? I ...
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1answer
88 views

Locii of vertices of parallelograms

$ABCD$ is a given parallelogram. What is the locus of the remaining vertices of a parallelogram equal in area to $ABCD$ having $AC$ as its diagonal? I came up with the answer that the remaining ...
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3answers
86 views

Having the number 59 for example find $x$ which $x^2$ is closer but lower to the 59. In this case is 7

I don't know how to name this but this is what I would need. ...
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3answers
159 views

Three Circles Meeting at One Point

We have three triples of points on the plane, that is, $X=\{x_1, x_2, x_3\}$, $Y=\{y_1, y_2, y_3 \}$, and $Z=\{z_1, z_2, z_3\}$, where $x_i, y_i, z_i$ are points on the plane. I was wondering if there ...
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3answers
203 views

Find the point that divides the line segment $3/5$ of the way

Given $A(-6,4)$ and $B(19,29)$, find the point that divides the line segment $AB $ $3/5$ of the way from $A$ to $B$. Since the line is not horizontal or vertical, it is a little more difficult. I ...
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1answer
186 views

The normal of a surface that passes through the origin

this is my first question here. Suppose I have a surface as follow: $$x^2+y^2+z^2=9$$ The gradient of the surface at a particular point $P = (x_0,y_0,z_0)$ is just $(2x_0,2y_0,2z_0)$ and the ...
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2answers
76 views

Find the region at which the curve transitions from linear growth to exponential growth

I have a set of $2D$ points: let's say $x$ and $y$. I start from $x = 0$ and increment by $1$ and for each increment I record the value of $y$. So, $y$ is a function in $x$. If i plot the graph of $x$ ...
1
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1answer
150 views

find the height of the tower

A person standing at a point $A$ finds the angle of elevation of a nearby tower to be $60^{\circ}$. From A, the person walks a distance of $100 ft$ to a point $B$ and then walks again to another point ...
2
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0answers
82 views

Locus using Euclidean geometry

Let $P$ be any point in the plane. Find the locus of $P$ such that $PA^2 + PB^2 = PC^2$, where $ABC$ is a triangle. I have found the locus. It's a circle having center at point $Q$, such that $...
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1answer
216 views

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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2answers
281 views

Axis of rotation of composition of rotations (Artin's Algebra)

Say $R_1$, $R_2$ are rotations in $\mathbb{R}^3$ with axes and angles $(v_1,\theta_1), (v_2,\theta_2)$ respectively. Since $SO_3$ is a group, we have that $R_2 \circ R_1$ is a rotation with some axis $...
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2answers
266 views

Closest Point on a Sphere to Another Point

Given a sphere $S(c,r)$, $c$ being the center point $(x,y,z)$ and $r$ being the radius, there is a point $p(x', y', z')$ which is either inside or outside $S$. I want to find the point $q$ such that $...
0
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1answer
65 views

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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0answers
52 views

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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3answers
126 views

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 ...
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2answers
314 views

Projection matrix to project a point in a plane

How to determinate the 4x4 S matrix so that the P gets projected into Q, on the XZ (Y=0) plane? Q = S P
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2answers
46 views

Geometry, two perpendicular lines

"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 ...