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

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4
votes
0answers
17 views

Given two points, how to find a circle through them that's also tangent to the $x$-axis?

A seemingly simple geometry problem that is surprisingly difficult. I want to find the radius of a circle that is tangent to the $x$-axis, but also must contain two given points. I understand there ...
0
votes
0answers
9 views

Vertical/Horizontal stretch ratio.

I'm trying to calculate 3 points on a grid by using 3 other points and calculating a ratio. I have the original values of all the points and was using the ratio (x1/x2) (y2/y1) for the points however ...
-1
votes
0answers
15 views

How to find the image of the line $y=ax$ from upper half plane to poincare disk?

with cayley transformation $m(z)=\frac{z-i}{z+i}$ i cant find a solution for this exercise so if you have any suggestion for the solution will be very helpful ...
0
votes
0answers
10 views

To a given straight line in a given rectilinear angle, to apply a parallelogram equal to a given triangle.

There's again one small detail on which I'm not sure. (Proposition 44 - book 1) http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI44.html Here's the quote : "Then HLKF is a parallelogram, HK ...
-5
votes
0answers
23 views

Prove that ABCD is cyclic if and only if it is a rectangle [on hold]

Prove that $ABCD$ is cyclic if and only if it is a rectangle, in which case its circumcenter is the point where its diagonals intersects.
1
vote
0answers
12 views

min and max number of hexagons in hexagonal tiling

Is there a way to calculate the maximum and minimum number of hexagons in a hexagonal tiling of a surface with regular identical size hexagons, knowing the area of the surface and the area of the ...
0
votes
1answer
22 views

Speed at which hands of clock approaching one another.

A clock's hour hand's length is $1$, and its minute hand's length is $r$. First I had to find the distance between the tips of the hands at 4:00. I did this using the law of cosines. This gives me ...
0
votes
4answers
32 views

For every integer vector $\overrightarrow{a}$,there is a integer vector $\overrightarrow{b}$ such that $\overrightarrow{a}\bot\overrightarrow{b}$

In $R^3$,show that for every integer vector $\overrightarrow{a}$,there is a integer vector $\overrightarrow{b}$ such that $\overrightarrow{a}\bot\overrightarrow{b}$ Generally,in $R^n$,for every ...
0
votes
0answers
11 views

hexagonal tessellation (tiling): uniform distribution of centers of hexagons?

Consider a disk of Radius $R$. We divide the disk into n equal sectors (in the form of pizza slices) . $n= 2^i$ and $i$ is a non-negative integer. Each sector is enclosed with two radii and an arc ...
-3
votes
1answer
26 views

How do you express the following equations for a circle?

A circle of radius a is centered at a point r1. (a) Write out the algebraic equation for the circle. (b) Write out a vector equation for the same circle. (c) How would you modify (a) and (b) above ...
6
votes
4answers
65 views

Prove that for any given $c_1,c_2,c_3\in \mathbb{Z}$,the equations set has integral solution.

$$ \left\{ \begin{aligned} c_1 & = a_2b_3-b_2a_3 \\ c_2 & = a_3b_1-b_3a_1 \\ c_3 & = a_1b_2-b_1a_2 \end{aligned} \right. $$ $c_1,c_2,c_3\in \mathbb{Z}$ is given,prove that $\exists ...
1
vote
4answers
35 views

Find the length of chord $BC$.

On a semicircle with diameter $AD$. Chord $BC$ is parallel to the diameter.Further each of the chords $AB$ and $CD$ has length of $2$ cm while $AD$ has the length $8$ cm.Find the length of $BC$. ...
-1
votes
1answer
26 views

Geometry volume and surface area [on hold]

How do I get the two sides of the top of the prism ($9$ & $9\sqrt{3}$)
0
votes
1answer
20 views

What is the optimal way to detect a collision between an AABB figure and a non-AABB figure?

Background I'm looking to do this programmatically in Java, but if desired you can post solutions in C/C++ or plain English instructions if you're not a programmer, but I would appreciate an ...
0
votes
3answers
47 views

If the surface area of a box is 32 and its volume is doubled what is the new surface area? [on hold]

Original surface area :32 Original volume: x New volume: 2x What is the new surface area? Please provide an explanation or show work, I don't know how to do it.
0
votes
1answer
23 views

How can I derive the resultant of 2 bearing/elevation pairs

Say, for example I have a gimballed camera mounted on a metal plate, which is itself fixed horizontally to a boat. I can measure the elevation and bearing of both the camera with respect to the plate ...
1
vote
2answers
23 views

What does it mean for a set of closed shapes to intersect?

To my understanding, a "shape" is a set of points in $n$-dimensional space. e.g., rectangles, triangles, lines, spheres, hyperspheres, etc.. For two (or any amount) of shapes to "intersect", the ...
2
votes
1answer
15 views

Find the three closest surrounding neighbors from a data

I have a data of coordinates $x$ and $y$ where we know the range of both variables, e.g. $(x,y)\in[0,1]^2$. So for a given any random point $\theta_0=(x_0~~y_0)^T$ in the range of $x$ and $y$ I would ...
3
votes
4answers
93 views

How to prove that a straight line is an infinite set of points?

From the basic elementary level when we start reading geometry we get this idea developed in us that a straight line is the conjuction of infinite points.but how to prove this? I mean is this an ...
1
vote
2answers
21 views

Find the ratio of curved surface area of frustum to the cone.

In the figure, there is a cone which is being cut and extracted in three segments having heights $h_1,h_2$ and $h_3$ and the radius of their bases $1$ cm, $2$cm and $3cm$, then The ratio of the ...
0
votes
1answer
17 views

Deriving the Geodesic Equation

I found a derivation of the geodesic equation that includes this step as I write it: $$ \frac{d (g_{ab}\dot{x}^b)}{dt}=\frac{1}{2}\partial_ag_{bc}\dot{x}^b\dot{x}^c \Rightarrow \\ \\ ...
8
votes
0answers
44 views

An interesting property between a hyperbola & parabola

It is well known that when two tangents to a parabola are perpendicular to each other, they intersect on the directrix. In other words, the intersection point of the two tangents make a straight line, ...
1
vote
0answers
35 views

What is real $R$ so that every subset of Euclidean space with diameter one is inside a ball of radius $R$?

What is the real number $R$ so that for every $n$ every $S \subseteq \mathbb{E}^n$, for which $d(S) = \sup\{|x-x'| \mid x,x' \in S \} = 1$, is inside some closed $n$-ball of radius $R$? In particular, ...
2
votes
5answers
357 views

Creative way to find this area

Let's say We have a circle with center at $(0,0)$ with radius $r$ and we have the line $y=a$ where $0 \leq a \leq r$. the question is what is the area that between the circle and the line $y=a$(the ...
1
vote
0answers
20 views

4 faces sharing an edge. Which pairs of faces belong together.

I'm sorry for the somewhat vague title, but I'm not sure how to better describe the problem. I have a single solid in 3D (arbitrary, closed collection of faces). This solid internally has a part ...
1
vote
1answer
24 views

Elliptical section of a right circular cone [on hold]

A right circular cone, having cone angle $\alpha=40^o$, is thoroughly cut with a smooth plane (normal to the plane of paper as shown by the produced line AB in the diagram below) making at an acute ...
2
votes
1answer
56 views

BMO1 2006/07 Question 4 Geometry Problem

$4.$ Two touching circles $S$ and $T$ share a common tangent which meets $S$ at $A$ and $T$ at $B$. Let $AP$ be a diameter of $S$ and let the tangent from $P$ to $T$ touch it at $Q$. Show that $AP = ...
1
vote
0answers
17 views

Hexagonal tessellation: tesselating a pizza slice shape sector of a circle with a special constraint

Consider a sector(in the shape of a pizza slice) of a disk of radius $R$ such that the sector is enclosed by two raddi and and an arc, where the arc subtends an angle $\frac{2\pi}{n}$. We tessellate ...
0
votes
1answer
12 views

Finding the coordinate at time $t$ of a line determined by the points $(x1,y1), (x2,y2)$

I have the problem here, I create a program that clipping a line with the input (x1,y1,x2,y2). but the algorithm only explain until I get ...
3
votes
0answers
23 views

How to state Pythagorean theorem in a neutral synthetic geometry?

In some lists of statements equivalent to the parallel postulate (such as Which statements are equivalent to the parallel postulate?), one can find the Pythagorean theorem. To prove this equivalence ...
0
votes
0answers
30 views

hexagonal tesselation symmetry

Consider a disk of radius R. We tessellate the surface of the disk using hexagonal tessellation where each hexagon has a circumradius r. The tessellation is made such that the center of one of the ...
0
votes
1answer
14 views

Test if a vector is pointing towards the center of an ellipse

I have an ellipse : $$x = h + a\cos t \cos\theta - b\sin t \sin\theta \\ y = k + b\sin t \cos\theta - a\cos t \sin\theta$$ Let's say if we have a normal vector $n$ to the ellipse, on a point $p$ ...
0
votes
0answers
20 views

Triangles which are on the same base and in the same parallels equal one another.

I have a small question regarding proposition 37 of the elements of Euclid. http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI37.html The only problem I got with the proof is the fact that we ...
0
votes
0answers
19 views

Getting the normal vector of a point on an ellipse

I have a rotated (by $\theta$) and translated (by $h,k$) ellipse given by: $$ x = h + a\cos t \cos\theta - b\sin t \sin\theta \\ y = k + b\sin t \cos\theta + a\cos t \sin\theta $$ By the normal ...
1
vote
0answers
15 views

Find the minimum total surface area of the cylinder in given circumstances.

Six solid hemispherical balls have to arranged one upon the other vertically .Find the minimum total surface area of the cylinder in which the hemispherical balls can be arranged, if the radii of ...
1
vote
4answers
30 views

what is the volume of cylinder if

The total surface area of a cylinder is $80\pi~\text{cm}^2$ and the difference between the height and the radius is $2~\text{cm}$. What is the volume of that cylinder? I have tried to find the ...
0
votes
2answers
60 views

Triangle whose side lengths and area are rational numbers [on hold]

Does there exist a triangle with side lengths given by rational numbers $x$, $2x$, and $y$ such that the triangle's area is also rational number?
0
votes
3answers
55 views

Prove that $\alpha$ lies between $0$ and $4$.

Let $a,b,c$ be the length of the sides of the triangle $ABC$ . Given $(a+b+c)(b+c-a)=\alpha bc$.Then Prove that the value of $\alpha$ lies in between $0$ and $4$. ...
3
votes
2answers
81 views

BMO1 2006/07 Question 2 Geometry Problem

$2.$ In the convex quadrilateral $ABCD$, points $M,N$ lie on the side $AB$ such that $AM = MN = NB$, and points $P,Q$ lie on the side $CD$ such that $CP = PQ = QD$. Prove that Area of $AMCP=$ Area of ...
3
votes
2answers
33 views

Total area for a natural nested set of convex polygons.

Suppose we have a convex polygon $P_0$ with $n$ given vertices, and we want to "nest" polygons $P_j$ for $j > 0$ by taking the midpoints between edges of $P_{j-1}$ as the vertices. For a regular ...
2
votes
2answers
43 views

Find the length of tangent $x$.

Two circles $C_1$ and $C_2$ of radius $2$ and $3$ respectively touch each other as shown in the figure .If $AD$ and $BD$ are tangents then the length of $BD$ is $a.)3\sqrt6\\ b.)5\sqrt6\\ ...
0
votes
0answers
13 views

Straight lines which join the ends of equal and parallel straight lines in the same directions are themselves equal and parallel

I have a small question regarding proposition 33 of the elements of Euclid. http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI33.html We want to prove that two lines joining equal parallels ...
1
vote
2answers
27 views

ARML: Tangent congruent circles forming a right circular cone

Four congruent circles are tangent to each other and tangent to the edges of a sector as shown. If the straight edges are joined to form a right circular cone with the vertex at P, the radius ...
-2
votes
2answers
36 views

Geometry/ trig. [on hold]

I need to cut an angle brace (hypotenuse)21 inches, between two shelves 18 inches apart, shelf to be braced is 8 inches wide What are the degrees of a right triangle, with sides 18", 21", 8" ?
0
votes
1answer
15 views

Action of the Euclidean group, generalizing linearity?

I have a vector $v \in \mathbb{R}^2$ and two elements $(A,a)$ and $(B,b)$ of the Euclidean group $E(2)$. If the relation $$[(A,a)(B,b)](v) = v$$ holds, can I say that $(A,a)(B,b)$ is the neutral ...
0
votes
1answer
30 views

Probability of points in a triangle

$O(2,3)$, $A(2,0)$, $B\left(1,\dfrac{1}{\sqrt{3}}\right)$ are the vertices of $\Delta{OAB}$ on the $\text{x-y}$ plane. Let $\text{R}$ be the region consisting of all points $P$ inside the triangle, ...
-2
votes
4answers
50 views

Find Perimeter of shaded region in semicircle. [on hold]

What is the Perimeter of shaded region in semicircle if four small semicircles have radii of 1,2,3,4 respectively? a. 10 $\pi$ b. 20 $\pi$ c. 40 $\pi$ d. 60 $\pi$
0
votes
1answer
42 views

Can you determine the volume of this pyramid?

A coworker posted this image on the whiteboard today: And the quote is "Can you determine the volume of this pyramid?" Now, I've been able to determine the following: And ∠BDC is a right angle. ...
0
votes
1answer
20 views

How to solve geometry question on internal tangency

Let $\Gamma_1$ be a circle with centre at the Point $O$ and radius $R$. Two other circles $\Gamma_2$ and $\Gamma_3$ with centres $O_2$ and $O_3$ respectively are internally tangent to $\Gamma_1$ and ...
1
vote
1answer
29 views

Relating the singular values of a scaled matrix to its determinant

Let $A$ be a real, $n\times n$,full rank matrix with singular values: $\sigma_1\ge\dots \ge \sigma_n$. Assuming the rows of $A$, $a_1,\dots,a_n$ are scaled so that $\|a_i\|_2 = 1$ for $i=1,\dots,n$, ...