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

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1answer
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Expected area of the union of two and three circles

I have done the part of two circles. Suppose that there are two intersecting circles with radius R. And let the distance between the center of two circles is D.(0$\le$D$\le$2R) The intersection of ...
3
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0answers
20 views

Can three diagonals in a $2k+1$-gon intersect?

Is it possibly to find three diagonals in a regular $2k+1$-gon that intersect? More particularly can they intersect inside the polygon? I am looking for an elementary solution. Can one be found ...
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1answer
33 views

Minimizing the area of the triangles containing a square of side $1$

This exercise is from a past admission exam to an Italian institute: Among all the triangles that contain a square of side $1$, which ones have minimum area? I have solved it, however I'd like ...
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0answers
14 views

How do I convert a line angle to a scale of [0,360]?

I have two points, (x1,y1) and (x2,y2), that I'd like to draw a line between. I know I can figure out the angle of that line using Sin(th)=opp/Hyp: ...
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2answers
37 views

Rotate a point on a circle with known radius and position

Having a circle $\circ A(x_a, y_a)$ of radius $R$ and a point on the circle $B(x_b, y_b)$, how can we rotate the point with a known angle $\alpha$ (radians or degrees, it doesn't really matter) on the ...
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4answers
35 views

Explanation of this integral's solution

While doodling around with circles and associated geometry, I've stumbled across this integral (I'm led to believe it is correct but I have not found or created any proof): ...
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0answers
38 views

projecting and mapping disparate coordinate systems

Edit: To say more simply, there a are two parts. 1. I have a 2D plane existing inside of a 3D coordinate system. The 2D plane is defined by two 3D vertices. How do I then transform coordinates that ...
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0answers
13 views

Rhombus and collinear points [on hold]

A rhombus ABCD is given.Arbitrary point E is chosen on the diagonal AC (E is different from A and C). Points N and M are selected on lines AB and BC,respectively, so AE=NE and CE=ME. K is intersection ...
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1answer
22 views

triangle inequality theorem

triangle inequality theorem states Sum of any $2$ sides must be greater than $3$rd side. I am doing exercises in my text book. For example, I am checking if $10,3,5$ can form a triangle. $10+3 > ...
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3answers
53 views

Do 4 points in ${\mathbb R}^2$ in convex position define a unique elliplse that passes through those 4 points?

So it takes 3 distinct points in the plane, that are not collinear, to define a unique circle that passes through the points. So what about ellipses? Arguing naively in terms of degrees of freedom ...
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0answers
17 views

3D projection onto 2D plane to determine transformation matrix?

I'm not sure if there is an actual solution to this problem or not, but thought I would give it a shot here to see if anyone has any ideas. So here goes: I basically have three vertices of a rigid ...
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0answers
13 views

projecting different coordinate systems in 3D space

Suppose I have a local set of coordinates {i,j,k} in which there are points A,B,C,D,E,F,G. I do not know their absolute positions, but I know their relationshit to one another. This local coordinate ...
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1answer
45 views

Why does it matter that the group of rotations act *freely* in Tao's proof of the Hausdorff paradox?

Consider the following extract from this expository article: (This is a key step in proving the Banach-Tarski theorem.) My question is: if the action of $G$ on $S^2 - C$ were not free, would the ...
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0answers
19 views

Geometry Reflections Help [on hold]

Pentagon ABCDE and pentagon A'B'C'D'E' are shown on the coordinate plane below: Pentagon ABCDE and pentagon A prime B prime C prime D prime E prime on the coordinate plane with ordered pairs at A ...
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1answer
44 views
0
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1answer
54 views

Geometric Problem. Find an angle.

In△ABC , AB=AC ∠ACB=72∘. D is a point in △ABC such that ∠DBC=42∘,∠DCB=54∘.Find an angle∠BAD. Is there any geometric method?
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2answers
63 views

A simple area problem

I found this problem from a book (level = grade 7). In the attached figure, ABCD is a square with sides 10 units. Need to find the area of the quadrilateral PQRS built inside it as shown. The only ...
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0answers
14 views

tangential elliptical arc to two given ellipse [on hold]

How to draw using purely geometric methods. Given two ellipse and an elliptical arc touching the two ellipses at the end points of a line drawn connecting two focii of the two ellipse. I tried the ...
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2answers
25 views

Co - ordinates of a point lying on the perpendicular bisector of a segment.

Having two points $A(xa, ya)$ and $B(xb, yb)$ and knowing a value $k$ representing the length of a perpendicular segment in the middle of $[AB]$, how can I find the other point of the segment? ...
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1answer
30 views

Mapping from Poincare's disk model to UHP

I have a question that : How can I map any point in Poincare's disk model to Upper-half-plane model? I know the function $$f(z) = \frac{z + i}{iz+1}$$ But I want to know the geometric ...
1
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1answer
19 views

Find coordinates of a 2D plane within a 3D plane

I'm not sure this is the right place to ask this question, if not I do apologies and I will move on. I am asking this question as a programmer, however it seemed entirely maths based. Image one is ...
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1answer
23 views

Question on dimensions of a prism. [on hold]

The surface area of a right prism is 224 square feet, and the length of a side of the square base is one-third the height. What are the dimensions of the prism?
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2answers
38 views

What is the motivation behind this (tested, working) 2d coordinate transform?

I am working with programming and 2d geometry and need to transform between two different coordinate systems. I have two different representations of the same world, where I can sample any point and ...
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1answer
20 views

Given line $e$ and plane $\alpha$, find all points $Q$ on $e$ such $d(Q, P)= d(Q, \alpha)$

Can someone help me with this question and show my step by step process. I am unable to solve it. Thank you. $P(4,2,5)$ The plane $\alpha$ is given by $2x+y-2z=2$ The line $e$ is given by ...
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1answer
14 views

Radical Axis of Collinear Points.

If I have three collinear points in $\mathbb{R}^{2}$, such that define a three different circles that pass through a point, then this three circles can define the same radical axis?? I think this a ...
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0answers
15 views

Maximally symmetric discrete subsets of the sphere

Let $d\geq 1$ and $\mathbb{S}^{d}=\{x\in\mathbb{R}^{d+1}:|x|=1\}$. For $X\subset\mathbb{S}^{d}$, $|X|<\infty$, let $T(D)=\{A\in\mathbb{R}^{(d+1)\times (d+1)} : A\in \text{SO}(d+1), A(X)=X\}$. ...
5
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1answer
59 views

How to change a $9\times 16$ rectangle to $12\times 12$ square?

Given a $9\times 16$ sq. unit rectangle. You have one change, you can cut the $9\times 16$ sq. unit rectangle only once and join the two parts to get a square of dimension $12\times 12$ sq. unit. ...
2
votes
2answers
22 views

Closest point of parameterized curve has orthogonal position vector to tangent

Let $\alpha(t)$ be a parameterized curve which does not pass through the origin. If $\alpha(t_0)$ is a point of the trace of $\alpha$ closest to the origin and $\alpha'(t_0)\ne 0$, show that the ...
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2answers
44 views

Triangle $ABC$ is isosceles with $AB=AC$, and $D$ is the midpoint of $AB$. If $\angle{BCD}=\angle{BAC}=θ$, then $\cos θ$ equals…?

Triangle $ABC$ is isosceles with $AB=AC$, and $D$ is the midpoint of $AB$. If $\angle{BCD}=\angle{BAC}=θ$, then $\cos θ$ equals...? I was doing some UKMT past paper questions - this question is ...
1
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1answer
50 views

Why does the surface area of the hypersphere go to zero as the number of dimensions goes to infinity?

Why does the surface area of the hypersphere go to zero as the number of dimensions goes to infinity? Is there an intuitive reason?
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1answer
17 views

A sheet of cardboard measures 15cm by 7cm. Four equal squares are cut out of the corners and the sides are turned up to form an open box. [on hold]

a) If the edge of the cut out square is x cm, express the dimensions of the box in terms of x. b) What are the restrictions placed on the values of x? (i.e. the implied maximal domain).
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1answer
42 views

Quadratic forms and midpoints

The midpoint of the vectors $u$ and $v$ is $w=\frac{u+v}{2}$. In euclidean geometry, an alternative characteristic of midpoints is $|v-w|=|u-w|=\frac{1}{2}|u-v|$. I wonder if this generalizes to ...
0
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2answers
66 views

Compute the integral over the volume of a torus,

In $\mathbb R^3$, let $C$ be the circle in the $xy$-plane with radius $2$ and the origin as the center, i.e., $$C= \Big\{ \big(x,y,z\big) \in \mathbb R^3 \mid x^2+y^2=4, \ z=0\Big\}.$$ Let $\Omega$ ...
0
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1answer
36 views

Line segment equation in polar coordinates

I have a line segment given by two points $A$ and $B$. $$A+u(B-A), u\in[0,1]$$ when doing calculations with this segment, it would be advantageous to have it written in polar coordinates around some ...
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4answers
33 views

Slope of a line segment.

If $A(x_1, y_1)$ and $B(x_2, y_2)$, we know that slope $m = \frac {(y_2 - y_1)} {(x_2 - x_1)}$. What decision can we take aout the line segment when, $m = \frac 0 0$, $m = \frac {dy} 0$, and, $m = ...
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0answers
23 views

Find the constant $k$

In a graph $G$ with $n$ vertices, let $T_1$ be the number of triangles one can make and $T_2$ be the numbers of tetrahedrons. Find the least constant $k$ such that $(T_2)^3\le k.(T_1)^4$.
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3answers
40 views

Question about concyclic points on the coordinate axes

If the points where the lines $3x-2y-12=0$ and $x+ky+3=0$ intersect both the coordinate axes are concyclic,then the number of possible real values of k is (A)1 ...
4
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2answers
50 views

External bisectors of the angles of ABC triangle form a triangle $A_1B_1C_1$ and so on

If the external bisectors of the angles of the triangle ABC form a triangle $A_1B_1C_1$,if the external bisectors of the angles of the triangle $A_1B_1C_1$ form a triangle $A_2B_2C_2$,and so on,show ...
0
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0answers
64 views

Complex numbers: conjugate [on hold]

Can anyone help me with this? Let $z^*$ be the complex conjugate of $z$ (a) Show that $ (zw)^* = z^*w^*$ (b) Prove by induction that $\ (z^n)^*=(z^*)^n$ for all positive integers $n$.
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4answers
90 views

Proof of Pythagorean theorem without using geometry for a high school student?

There are some proofs of Pythagoras theorem which don't even require high school maths to understand, but they all are using shapes to prove of the theorem. However, I am trying to find some proofs of ...
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3answers
26 views

Finding the equation of a circle given 3 points without using elimination [on hold]

find the equation of the circle using points (-4,-4) (-3,1) (2,0) without using elimination.
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3answers
45 views

If a triangle has 2 sides of equal length, is it isosceles?

I know that if the 2 angles of a triangle are the same, it is isosceles. But what if the two sides are the same? Can we conclude that the corresponding angles are the same and it is isosceles?
2
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1answer
24 views

Creating Gosper Curve by geometry

I am reading a book by "Fractals, Chaos and Power Laws" by Manfred Schroeder.On page 13, it produces seven fractal tiles from seven hexagons by breaking up each side into a three-piece zigzag as shown ...
2
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0answers
52 views

IMC 2014, Problem 4 [Day 2]

We say that a subset of $\mathbb{R}^{n}$ is $k$-almost contained by a hyperplane if there are less than $k$ points in that set which do not belong to the hyperplane. We call a finite set of points ...
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0answers
23 views

How to derive the relation $T=S_1$ for the equation of a chord of an ellipse whose midpoint is given? [on hold]

How to derive the relation $T=S_1$ for the equation of a chord of an ellipse whose midpoint is given as ( x1, y1 ) ? where T = x(x1)/aa + y(y1)/bb - 1 and S1 = (x1)(x1)/aa + (y1)(y1)/bb - 1 where 2a ...
7
votes
1answer
110 views

A movement requires 1 dimension, a rotation requires 2 dimensions, a what requires three dimensions?

Movement A zero-dimensional object cannot move. A one-dimensional object can move in one dimension (the x-axis). A two-dimensional object can move in two dimensions (the x-axis and y-axis). A three ...
1
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1answer
48 views

Projection of the XY plane.

Is there a way to project the infinite Complex plane to either the Poincare disk or the unit disk - for all values of x + iy ?
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1answer
40 views

A formula to calculate the partial volume of a capsule or tank?

We are trying to ascertain the correct formula discussed in this post. The volume formula for a capsule (a cylinder with a hemisphere at both ends) is, $$V_c = \pi r^2 H + \frac{4}{3}\pi r^3\tag1$$ ...
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1answer
35 views

Find the value of $x$ below

$AB=DC$, Find the value of $x$ I tried with Law of Sines but I get different answer every time
4
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2answers
69 views

$D, E, F$ are respectively projection of $O$ on $BC, CA, AB$. Prove that $\cot{\angle ADB} + \cot{\angle BEC} + \cot{\angle CFA} =0$

Let $O$ be an arbitrary point located inside the triangle $ABC$. Let $D, E, F$ be (respectively) the projections of $O$ on $BC, CA, AB$. Prove that $$\cot{\angle ADB} + \cot{\angle BEC} + ...