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

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0answers
101 views

Finding midpoint of a cluster of points (2D)

Probably the dumbest question of the day, but given a cluster of points on a 2D graph, to find the "average" coordinate that sits at the midpoint of the cluster, is it as simple as averaging each ...
10
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3answers
376 views

Elementary proof that there is no paradoxical decomposition using triangular pieces

I am teaching a geometry course and I am trying to understand two definitions in the textbook ("Geometry with Geometry Explorer" by Michael Hvidsten.) Definition: The area of a rectangle is its base ...
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1answer
35 views

How to mesure lengths and areas in other euclidean, spherical and hyperbolic geometry?

I am learning how to measure lengths an areas in euclidean, spherical and hyperbolic geometry, but I'm getting very confused. First of all, I am told a rectifiable curve $\gamma$ has length defined ...
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3answers
421 views

Is convex hull of a finite set of points in $\mathbb R^2$ closed?

Is the convex hull of a finite set of points in $\mathbb R^2$ closed? Intuitively, yes. But not sure how to show that. Thanks!
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4answers
51 views

How do i prove this set has at most 2 elements?

Let $w,\alpha\in\mathbb{C}$ and $\delta,\epsilon >0$ such that $(w,\delta)\neq (\alpha,\epsilon)$ Define $G=\{z\in\mathbb{C} : |z-\alpha|=\epsilon \text{ and } |z-w|=\delta\}$ How do i prove that ...
2
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2answers
231 views

Orthogonal tangents to an ellipse [duplicate]

This is the problem I found back in the first year in the university. Suppose we have a non-degenerate (i.e. not a point and not an empty set) ellipse $E\subset \Bbb R^2$. Now define a set $D$ by a ...
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1answer
41 views

What is the radius of the $12$ spheres surrounding one of radius $1$ if all touch each other?

A sphere of radius $1$ is surrounded by $12$ spheres of radius $1$. But a small gap is left. What is the radius of the upper layer of spheres if all the $12$ kiss each other so that no gap is left? I ...
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2answers
48 views

area of ​​a quadrilateral

Get the area of ​​a quadrilateral? $‎\angle ‎A‎‎‎_{1}‎+‎\angle ‎C‎_{3}‎=30‎^{‎\circ‎}‎‎‎‎‎$‎ $\angle ‎A‎‎‎_{2}‎+‎\angle ‎C‎_{4}‎=90‎^{‎\circ‎}‎‎‎$ $CD=9, DA=5, BC=8 , AB=4$
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1answer
26 views

Constructing a line given two other lines, two angles, and a distance.

Two non parallel lines $l$ and $m$ are given. For given two angles $A$ and $B$ we have to construct a line $n$ such that it makes angles $A$ and $B$ with lines $l$ and $m$ respectively. Line $n$ ...
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1answer
37 views

Why is this a line equation?

Define $$L=\{z\in\mathbb{C} : cz + \overline{cz} + w = 0\}$$ Where $c$ is a nonzero constant. How does $L$ represent a line?
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4answers
304 views

Equal perimeter and area

Find all triangles of which perimeter and area are numerically equal. I have got solution for right angle triangles but not of others
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3answers
38 views

Sketching coordinates of C when it meets the x and y axis

The question: Curve $C$ has the equation $y=(x-k)^2 (x-k+2)$ Where $k$ is a constant and $k > 2$. Sketch $C$, showing the coordinates of the point where $C$ meets the $x$ and $y$ axes. I am ...
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4answers
68 views

Mathematics based on triangles

How to find the third cordinate of a triangle , where as other two points are known. and a angle is known. Lets say , the two points are (0,0) , (600,0) and we need to find the third cordinate . ...
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1answer
50 views

How find this this distance $d_{1}d_{2}=b^2$

On the plane we have two points $A(\sqrt{a^2-b^2},0),B(-\sqrt{a^2-b^2},0)$ with $a>b>0$ and the line $L$, of which the equation is given ...
4
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1answer
235 views

How prove this result $\frac{x}{y}=\sqrt{\frac{\sqrt{5}+1}{2}}$

A tetrahedron $A-BCD$ is such all four faces are similar right triangle. and we let $$AB=a,BC=b,AC=c,AD=d,BD=e,CD=f$$ define $$x=\max{(a,b,c,d,e,f)},y=\min{(a,b,c,d,e,f)}$$ show that: ...
3
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1answer
162 views

Locus of the centres of equilateral triangles (contest problem)

Given a triangle $A_0A_1A_2$ determine the locus of the centres of the equilateral triangles $X_0X_1X_2$ satisfying the condition that each of the lines $X_kX_{k+1}$, $k=0,1,2$ passes through ...
15
votes
2answers
581 views

Can a cube always be fitted into the projection of a cube?

If we project the unit cube, i.e. a axis parallel cube with side length 1 centered at the origin, in $\mathbb{R}^n$ onto a $k$-dimensional subspace of $\mathbb{R}^n$ which contains the origin, can we ...
4
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3answers
3k views

Equation of a torus

First I am a newbie in maths so please forgive me if I am not as rigorous as you would like, but do not hesitate to correct me. I want to find the equation of a torus (I mean the process, not just ...
0
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4answers
189 views

Circumcircles of a trapezoid

I was just wondering, what types of trapezoids have circumcircles? I know one of them might be isoceles trapezoids, but are there any others?
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1answer
26 views

Is there another way to solve the value field of a parameter of an line.

Assume $P$ is a point in line $x+y=m$, where $m \in \Bbb{R}$. There are two points $A,B$ in circle $$x^2+y^2 = 10$$ such that $PA$ and $PB$ are tangent lines of the above circle. If line: $x+y=m$ has ...
4
votes
1answer
258 views

Incidence Geometry Proof

How do I show that the axioms of incidence geometry follow as theorems from the following axioms. 1) There exist exactly four lines. 2) Any two distinct lines are incident with exactly one point. ...
2
votes
1answer
175 views

Relation between mean width and diameter

Question: Let $A$ be a compact set in $\mathbb R^n$. Is it always true that $\text{mean-width}(A) \ge C \cdot \text{diam}(A)$ for some constant $C$ depending only on the dimension? If not, is it ...
0
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1answer
43 views

2D is to face as 3D is to?

Essentially, if a point is a zero-dimensional component of an object, a line is a one-dimensional component, and a face is a two-dimensional component, what is a three-dimensional component? If there ...
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3answers
82 views

Geometry: Using Pythagorean Theorem to find lengths of triangle

I am helping someone do their geometry homework, and wanted to make sure I am even doing it correctly. Below is the problem: Here is how I solved it: We know B is the center, so AB=BE. ...
2
votes
3answers
64 views

Geometry: Find the angle x

I am trying to help my little sister do her geometry and seem to have forgotten my basic math skills. Here are a couple she sent me: Any help would be great! Once I get back into the grove I ...
2
votes
2answers
70 views

A proof related to geometric mean.

Can anyone solve this problem?? If a square is inscribed into a right triangle in such a way that one side of the square lies on the hypotenuse, then this side is the geometric mean between the two ...
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0answers
66 views

What does it mean to compute a normal to a triangle in a “clockwise direction”

I am trying to understand how this works. I am given 3 points, each representing a vertex of a triangle. I must then "organise" the points and calculate the normal of the resulting triangle in a ...
0
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0answers
26 views

Explanation of ratio and constructing a triangle with given ratio.

can anyone explain about ratio in some easy way and solve this problem?? Construc a triangle, given the angle at the vertex, the base, and its ratio to one of the lateral sides?? I am taking a ...
1
vote
1answer
117 views

Prove that the curves of the family $v^3/u^2=k$ are geodesics on a surface

Prove that the curves of the family $v^3/u^2=k$ where $k$ is a constant are geodesics on a surface with the metric $$v^2 \, du^2-2uv \, du+2u^2 \, dv^2$$ where $u,v \gt 0$.
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1answer
40 views

Is the “smallest” connected graph with $n+k$ edges a trivial generalization of the “smallest” connected graph with $n-1+k$ edges?

Consider a set $P$ of $n$ points in the plane. Using $n-1+k$ line segments, $k\geq 0$, these points can be connected (i.e., the graph in which the points are the vertices and the line segments the ...
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1answer
993 views

Geometry/Programming- Draw An Equilateral Triangle Given One Point And A Desired Rotation

I feel this question has a stronger mathematical basis than strictly computer science. I am currently drawing an equilateral triangle given its center and its radius like so. I would like to ...
-1
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1answer
32 views

Co-ordinate Geometry Equation of a circle

Find the equation of the circle having the lines x+1=0 and x-3=0 as tangents and with its center lying on the line y=3 I don't know much maths so if you could tell me also how to get the co-ordinates ...
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0answers
55 views

Prove that the figure is a trapezoid…

Knowing that triangle $ABC$ is an arbitrary one, points $D$,$F$,$G$ are midpoints of respective sides of the triangle(as you see in the picture ), and $CE$ is the altitude, prove that $DG$ is ...
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2answers
69 views

Does the median make angles in the same proportion as the sides?

Till I remember I had studied this in the lower classes, but am not sure whether this is true or not. In the figure CD is a median. Does CD divide the angles 1 and 2 in the same ratio of the sides a ...
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2answers
32 views

Prove that the measure of the angle…

How can I prove that the measure of angle $EBC$ is $60$? Thank you!
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2answers
78 views

How do I justify that a second order cone is an intersection of half space

I am studying convex optimization right now, and the text book claims that a second order cone is a collection of intersections of half space $$K_n = \bigcap_{ u:\|u\|_2 \leq 1} \{(x,y) \in R^{n+1 } ...
1
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1answer
135 views

Formula for greatest cross section of regular dodecahedron

Is there a formula for the area of greatest cross section of a regular dodecahedron? For example, this can be viewed as finding a hole big enough for it to fit into.
1
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1answer
112 views

How to test for a polygon witn n vertices if it's nonintersecting polygon or not?

How can you design an algorithm to know if an n-vertex polygon nonintersecting ? On what criteria is the test going to be
1
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0answers
69 views

How to calculate normal (of magnitude 1) of a triangle?

I am currently doing a bit of geometry practice and wanted to know how to calculate the normal (of magnitude 1) of a triangle defined by 3 vertices: a, b and c`. ...
3
votes
3answers
789 views

Two triangles, only one different side, same area?

Suppose the following triangles: Where $BC = CD$. Obviously, the area of $\triangle ABC$ and $\triangle ACD$ are equal, since they both share the same base, and the same height, namely, $AB$. I ...
1
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1answer
65 views

Prove if one side of a triangle is a common measure of the other two sides, then the triangle is isosceles.

The definition of a common measure in my text book is this : A common measure of two segments is a third segment such that it is contained in each of the first two a whole number of times with no ...
1
vote
2answers
165 views

Finding the measure of an arc on a circle

If someone could work me through how to solve this, that would be great because I am stumped on this one. I know it looks like there is a lot of useless information in the picture, but there are ...
1
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1answer
84 views

Isometry in Euclidean space

The question is to show that an isometry from $\mathbb{E}^{1} \to \mathbb{E}^{1}$ is of the form $x \to ax + b $ from first principles, and determine the values $a$ can take. From my notes I know for ...
2
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1answer
127 views

Finding the equation of a plane in 3-D by using point-to-point distances

Assume that we have a plane $P(a,b,c,d)$ whose equation is unknown. We know that there is a point set $N = \{n_1, n_2, ...\}$ and $\forall n_i \in N$, $n_i$ is on $P$. Also, $\forall n_i, n_j \in N$, ...
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2answers
50 views

Provide a proof for the following problem…

How can I prove from this image that $BQ=2*PE.$ We know that $TM$ is parallel to $QD$ and that $CF$ is the bisector of angle $C$. As you see $QD$ and $TM$ are perpendicular to $CF$. Obiously I found ...
3
votes
5answers
499 views

Is the rhombic dodecahedron the only isohedral polyhedron that tiles 3-space (other than the cube)?

Is the rhombic dodecahedron the only face-transitive (or isohedral, i.e. all faces are the same) polyhedron that seamlessly tiles 3-dimensional Euclidean space (other than the cube)? I'm looking ...
0
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2answers
28 views

Proof required for the following problem…

Can you please give an explanation of how can I find that $K$ and $P$ are midpoints of respectively $CR$ and $CS$. NOTE! that: $AE$ and $DB$ are bisectors of angles $A$ and $B$. and $CK$ and $CP$ ...
0
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1answer
113 views

A quadrilateral ABCD satisfies the following relationship with respect to any point $M$ in the plane $AM^2+ CM^2 = BM^2 + DM^2$

This quadrilateral could be A parallelogram A rectangle A square A rhombus None of the above The answer can have multiple answers. Please provide the proof of your answer. I would be grateful ...
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2answers
32 views

Multiplying a vector by a constant

This is probably a very silly question, but I just can't remember... If $\vec{u}=-8i+32j$, how can I multiply it by a constant $a$? Would the new vector be: $\vec{u}=-8ai+32j$ ?
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1answer
61 views

Find the length of…

Find the length of $\overline{AD}$ knowing that it is divided into three equal parts by to tangent circles with radius respectively $3\sqrt{3}$ and $\sqrt{3}$ . Here's the graph: so the segments ...