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
10 views

Areas of triangles in hexagon

A hexagon $ABCDEF$ with parallel opposite sides is given. Prove that $[ACE]=[BDF].$ (here $[]$ denotes area of triangle) Since the sides are parallel does that mean that it is equiangular as ...
7
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6answers
198 views

Length of a Chord of a circle

I was wondering about the possible values that the length of a chord of a circle can take. The Length of a chord is always greater than or equal to 0 and smaller ...
0
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1answer
18 views

Geometry-||gm proof

$ABCD$ is a parallelogram in which $P$ and $Q$ are the mid points of the sides $AD$ and $BC$ respectively. If $BP$ & $QD$ intersect the diagonal $AC$ at $X$ and $Y$ respectively then prove that ...
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1answer
20 views

Find both the diagonals with area and side given [on hold]

Side of rhombus $= 65$ cm Area of rhombus $= 1024$ cm$^2$ Find the diagonals of the rhombus.
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1answer
25 views

Rotation addition with quaternions

My task is: "Describe rotation $S \circ R$ by axis and angle, where $R$ is rotation around $(0,1,1)$ by 90 degrees, and $S$ is rotation around $(1,-1,0)$ by 90 degrees." I should use quaternion ...
0
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0answers
23 views

Prove that $x+y \tan {\frac{(\alpha + \beta)}{2}}=0$ [on hold]

The line joining $A(a\cos{\alpha},b\sin{\alpha})$ and $B(a\cos{\beta},b\sin{\beta})$ is produced to the point $M(x,y)$ such that $AM: BM=b:a$; prove that $x+y \tan {\frac{(\alpha + \beta)}{2}}=0$ ...
0
votes
1answer
28 views

If the coordinates of the vertices $B$ and $D$ are $(7,3)$ and $(2,6)$ respectively, find the coordinates of the vertices $A$ and $C$.

The sides of the rectangle $ABCD$ are parallel to the co-ordinate axes. If the coordinates of the vertices $B$ and $D$ are $(7,3)$ and $(2,6)$ respectively, find the coordinates of the vertices $A$ ...
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0answers
12 views

How to visualize the product of two segments lengths? [duplicate]

So there seems to be easy ways to visualize addition. If three points a, b, and c are on the same straight line respectively, we can say that the sum of the lengths of ...
2
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5answers
54 views

How limiting/ heavy is the “triangle inequality” assumption?

Suppose a theorem proves something about a family of distance measures, with this the triangle inequality assumption. How limiting this assumption is in reality? What are some real-world examples of ...
0
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0answers
11 views

Translate and Rotate mesh

I have a mesh constituted of some vertices in 3d space, let's call them $(x_1,y_1,z_1),(x_2,y_2,z_2),\cdots,(x_n,y_n,z_n)$. The mesh's central point is $(0,0,0)$. How to find out the new coordinates ...
0
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1answer
18 views

The Name of a Polyhedron with 6 Quadrilateral Faces, 8 Vertices, and 12 Edges

(Don't say 'cube' or 'rectangular prism') I'm looking for a generic name for polyhedra with 6 Faces, 8 Vertices, and 12 Edges where each face could be any quadrilateral shape: rectangle, rhombus, ...
2
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1answer
18 views

Continous family of $n$-gons

The statement of this problem asks to show that if $A$ and $B$ are two distinct convex $n$-gons there is a continous family of convex $n$-gons such that the first in that family is $A$ and the last is ...
1
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0answers
9 views

Two new conics associated with two triangles inscribed a conic

1-Let $ABC$ and $A'B'C'$ be two triangle inscribed a conic. Let $P$ be a point on the conic. $PA$ meets $B'C'$ at $A_1$, define $B_1, C_1$ cyclically. $PA'$ meets $BC$ at $A'_1$ define $B'_1, C'_1$ ...
2
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0answers
15 views

Bound on “width” of points in a plane

Suppose we define width $w(P)$ of point set $P$ in a plane to be the ratio of the maximum distance to the minimum distance between the points in $P$. (Assume unique coordinates so that $w(p)$ is ...
0
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0answers
17 views

How to prove that a convex polygon is cyclic

Is there an easy way to tell if a convex polygon is cyclic? I was told that if the vertices of the $n$-gon are $A_1,A_2,\ldots,A_n$, it is enough to prove that $A_1A_2A_3A_i$ is cyclic for each ...
0
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0answers
18 views

Using oblique projection can you always rotate a triangle to look like an equilateral triangle? [duplicate]

Starting with any triangle using oblique projection, can you view any shape triangle from an angle to see it as an equilateral triangle?
0
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0answers
18 views

Regular/Right hyperbolas through three points

Find all equations of regular hyperbolas passing through the points $A(\alpha,0),B(\beta,0),$ and $C(0,\gamma)$. Attempt: I am assuming here that regular means right. We can then write the ...
0
votes
1answer
10 views

Having trouble drawing an arc using atan2 and specified range of arc.

I'm writing code. My arc invariant is: $$ \theta_0, \theta_1 \in [-2\pi, 2\pi], \\ \theta_1 - \theta_0 \leq 2\pi $$ where $\theta_0$ is the arc beginning angle and $\theta_1$ is the arc end angle. ...
0
votes
1answer
10 views

scalene trapezoid point of diagonals intersection

We have a scalene trapezoid. We know AB and CD bases and the diagonal AC. Be P the point of intersection of the two diagonals. Is it possible to find the general expression for AP?
0
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0answers
12 views

How to rotate a 3D object, using only local x-, y-, and z-rotations, so that it always faces a camera at the origin

I have been struggling with a difficult problem involving 3D rotations. I first came across this problem in a computer science context, but I've attempted to generalize it a bit before posting. (I ...
5
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1answer
46 views

$\pi$ is dependent on properties of geometry, assuming that we define it as $C/d$. Could then the $\pi$ also be an integer?

$\pi$ is dependent on properties of geometry, assuming that we define it as $C/d$. Could there be a geometry where $\pi$ is a rational number or an integer?
2
votes
1answer
32 views

Lebesgue measure and similarities

There is a well-known theorem in Euclidean geometry (Eucl. VI-19) that says that the ratio of the areas of any two similar polygons is equal to the square of the corresponding ratio of similarity. ...
1
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0answers
28 views

How do I group points into several circles with a given radius?

There is a set of points on the Cartesian coordinate system. How do I group points into several circles with a given radius and meeting the following requirements: each point belongs to a group at ...
0
votes
0answers
20 views

Line postulate in Geometry

It is a postulate that it is possible to draw a straight line from any point to any point. Firstly, if the points are the same does this postulate still hold? Secondly, if this is a fact that can't be ...
0
votes
2answers
35 views

Question on Geometry and cyclic circles

Two circles intersect at $P$ and $Q$.Through $P$ two lines $APB$ and $CPD$ are drawn to intersect circles at $A,B,C,D$. $AC$ and $DB$ when produced meet at $O$. How do I prove that $OAQB$ is cyclic ...
0
votes
1answer
33 views

Find the height $h$ of a circular segment based on the Radius $R$ and length $c$

I have found the formula for calculating the R of a circle, based on a circular segment, which is: $$ R=\frac{h}2+\frac{c^2} {8h}$$ where $R$ is radius, $c$ is the length of the segment, and $h$ is ...
0
votes
1answer
33 views

disk-disk intersection area

I have two disks of radii $R_1, R_2$ with distance between centers, $d < R_1 + R_2$. How can I find the surface area common to the two disks? Rationale: Solar irradiation / energy input in ...
0
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0answers
24 views

approximate projection into eigenvector space

Given a matrix A, $3 \times 3$, that is symmetric with zero diagonal, I calculate a matrix V, $3 \times 3$, whose columns are the corresponding right eigenvectors and a diagonal matrix D, $3 \times ...
-2
votes
0answers
22 views

The volume of a specific rectangular prism is represented by $V(x) = -2x^3 + 10x^2 + 300x$. How do roots, vertices, and end behavior apply?

The volume of a specific rectangular prism is represented by $V(x) = -2x^3 + 10x^2 + 300x$, where $x$ is the height of the prism. How do roots, vertices, and end behavior apply? How is the graph ...
2
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1answer
32 views

Jordan curve of infinite length

I was thinking about Jordan curve with infinite length and Koch snowflake seems to be a valid answer intutively. Can anyone give mathematical proof for this?
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2answers
26 views

Find the projection of the line $x+y+z-3=0=2x+3y+4z-6$ on the plane $z=0$

Find the projection of the line $x+y+z-3=0=2x+3y+4z-6$ on the plane $z=0$ The equation represents the line of intersection of two planes. Using augmented matrix $$ \begin{bmatrix} 1 & 1 ...
1
vote
3answers
37 views

How to prove that $x^2 + 3y^2 = 1$ is contained inside of the unit ball?

What is the best way to show that $S = \{(x,y) | x^2 + 3y^2 = 1\}$ is contained in the unit ball without graphing the set?
0
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1answer
33 views

How many points among them should we take to ensure that some two of them are less than the distance $1/5$ apart?

We are given a fixed point on a circle of radius $1$, and going from this point along the circumference in the positive direction on curved distances $0,1,2,\ldots$ from it we obtain points with ...
0
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0answers
12 views

Hyperplane of an mn-dimensional space [on hold]

Can someone explain to me why the hyperplane of a $mn$-dimensional space would have dimension $(m-1)n$?
1
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0answers
30 views

Given $5$ points on a sphere, divide the surface into $5$ congruent connected regions containing one point.

There are $5$ points on the surface of a sphere. Is it always possible to divide the surface into $5$ connected congruent regions such that each region contains one of the $5$ points?
0
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1answer
9 views

Computation of minimal axis-aligned bounding box of an arc segment.

I'm trying to compute the minimal bounding box of an arc segment so when it's time to render it, I only have to examine pixel coordinates within a minimal rectangular region. The code below covers ...
0
votes
1answer
22 views

Constructible numbers defined over the rationals

If $z$ is constructible, then its minimal irreducible polynomial has a degree a power of $2$. Does the polynomial have to be defined over the rationals? I am asking this because we can ...
0
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0answers
35 views

Having trouble interpreting the geometry of this setup.

A circular conductor, with cross section given by $(x-d)^2+y^2=b^2$, i.e. radius $b$ and centered on $x=d$, has a circular core, made up of the interior of the circle $x^2+y^2=a^2$, with ...
0
votes
1answer
22 views

What does equally oriented mean

What does it mean for two triangles to be equally oriented? I have heard this term a lot but I haven't seen a definition of it. I know that in $3$-space two triangles are considered to be equally ...
0
votes
1answer
11 views

Modulo angle (rotation) in $2D$ space

Input parameters: space dividor (number), vector (vec2) Desired result: Divide space in $X$ sectors, then move all vectors to one sector. (Angle of any vector wont be larger then $360/X$.) Example ...
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3answers
31 views

calculate radius of circle that by given length of square that is inside it

in this picture a length of square edge is 8 cm. I want to calculate the radius of circle. i try to calculate it, but i don't know how. I calculate this:
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vote
1answer
33 views

Showing that $\alpha$ satisfies the equation $\sin 2x=x$

This is an A level question. For better understanding, I will attach a screenshot of the question and the mark scheme. Question: Here's what I have done: $$A(OBA) = \frac 12r^2α$$ [basic ...
2
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1answer
46 views

Quarter Circle packing

Just today, I was making tortilla chips, and I began to wonder, what is the most efficient way to pack circular quarters onto the plane? This sort of circle packing is most efficient for circles, ...
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votes
2answers
72 views

The area of square [on hold]

What is the area of the square versus a,b and c ? Thanks Edit to clarify the question, based on the OP's response to comments. The segments with lengths $a$ and $c$ are parallel, joined by $b$ ...
0
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1answer
25 views

How do you calculate the change in thickness of a cylinder, if you shave off a flat section?

I have a piece of steel, cylindrical (hollow), 200mm outside diameter with 160mm inside diameter (...
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votes
2answers
33 views

How to determine that the 3 points given in homogeneous coordinates are collinear? [on hold]

How do I prove that the 3 points given in homogeneous coordinates are collinear? $$A=(1,3,2)^T, B=(0,6,8)^T, C=(3,3,-2)^T$$
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1answer
37 views

length of radius of circles between their tangents

In this question, we have five circle that touch each other. we draw their tangents. If we know that smallest circle radius is 8 and biggest circle radius is 18, then what is the length of PF? Note: ...
0
votes
1answer
18 views

Translate a Rectangle Position from 1 Image to another [on hold]

I have a Large Size Image.Since its too large for processing within a small time, i need to resize it.I have the coordinates of a rectangle in the resized image.Is there a way i can translate this ...
0
votes
3answers
22 views

Area of rectangle and triangle derivation

I was wondering about the derivation for the area of a triangle and the area of a rectangle. Of course, we all know them to be $\dfrac{1}{2}bh$ and $bh$ respectively, but where is the derivation of ...
0
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0answers
21 views

locus of a variable straight line [on hold]

Geometry: A variable straight line always intersects the lines x=c,y=0; y=c,z=0; z=c,x=0. find the equation to its locus. taking the equation of a line in parametric form and substitute the given ...