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
21 views

Brownian Motion in Confined space, any results?

I am searching for work regarding Brownian motion in a confined space, like a sphere or a cylinder, where the wall will serve as reflection boundary. I am wondering if it is possible to derive results ...
0
votes
2answers
49 views

Equilateral triangles

Let $ABC$ be a triangle with $AB = 1$, $AC = 2$ and $m(\widehat{BAC}) = 30^\circ$. We build on the outside the equilateral triangles $ABM$ and $ACN$. Let $D$, $E$ and $F$ be the midpoints of $AM$, $...
1
vote
2answers
30 views

Another method of finding area of hypocycloids

I was finding the are the of hypocycloids. Then it struck me that apart from integration, there could be another method of finding the area of the hypocycloid with different curves. But the problem is ...
-4
votes
1answer
41 views

What formulas can I use that can help me solve this? [closed]

A ferris wheel is elevated 1 m above the ground. When the car reaches its highest point on the ferris wheel, its altitude from the ground level is 31 m. How far away from the center, horizontally, ...
1
vote
2answers
27 views

Proving the equation of pair of lines is $x^2+2xy\sec(2\alpha)+y^2$

Prove that the equation of the pair of lines having an angle $\alpha$ with $x+y=0$ and passing through origin is $x^2+2xy\sec(2\alpha)+y^2=0$ I think we have to use rotation matrix but can't think of ...
2
votes
0answers
92 views

Number of ways to connect vertices of n squares with line segments

What is the number of ways to connect the vertices of n squares with non-intersecting line segments ? These line segments should not cross the edges of the given squares as well. Obviously $N(3)$ is ...
-2
votes
0answers
11 views

How to draw diagrams on math.stackexchange [migrated]

I am asking this question here because I don't have enough reputation to ask it on meta.math.stackexchange. How do I draw diagrams to answer questions on trigonometry and geometry. I mean something ...
0
votes
0answers
28 views

Rate of change in degree for archimedean spirals

(Apologies in advance if I don't use the proper math terminology.) I am trying to figure out the constant rate of change in degree for an Archimedean spiral of one rotation. If the "outside circle" ...
2
votes
3answers
42 views

What happens when $r \to \infty$? Will it be a line? (partial circle)

Let $a$ be a arc of particle circles, which is constant. What happens when $r \to \infty$? Will it be a line? Radius of partial circle : $r$, Arc of partial circle : $a$ and constant, For $r=r_0$ ...
3
votes
2answers
84 views

circle tangent to three circles

To-day I want to look at CCC - one circle tangent to three circles whose radii and positions of their centers are known. How does one solve this.. old fashioned ways like ruler and compass, or ...
0
votes
1answer
45 views

Problem on Equilateral Triangle and points

Equilateral $\triangle{ABC}$ with sides $2\sqrt{3}$. Let $P$ be the point outside$\triangle{ABC}$ such that points $A$ and $P$ lie opposite to $BC$. Let $PD$, $PE$, $PF$ be the perpendicular dropped ...
1
vote
1answer
22 views

Tracing the sides of an equilateral triangle

Is there any way I can get the points in 2D plane on the sides of an equilateral triangle for certain infinite animation sequence? For example in case of tracing the circumference of the circle, I ...
0
votes
0answers
21 views

Question about preserving cross ratios in projective geometry

Let $ABCDEF$ be a cyclic hexagon, such that $AF,BE,CD$ concur. Prove that $(F,D;E,C)=(A,C;B,D)$. I'm relatively new to projective geometry. This problem would be solved by perspectivity through the ...
0
votes
2answers
39 views

In the given figure, $PC$ is the angle…

In the given figure, $PC$ is the angle bisector of $\angle APB$ then prove that $XY||AB$ My attempt $$\angle APC=\angle BPC$$ $$\angle APC=\angle AQC$$ $$\angle XPB=\angle XQY$$ $P,X,Y$ and $Q$ ...
5
votes
0answers
148 views
+50

solve this 1999 problem with geometry

if $\bigodot P\bigcap \bigodot Q=A,B$,and the common tangent is $C,D$,and $E\in BA$,and $EC\bigcap \bigodot P=F,ED\bigcap \bigodot Q=G$,and if $\angle FAH=\angle HAG$ show that $$\angle FCH=\...
1
vote
3answers
42 views

Can we prove that $AD||PQ$ in the figure?

In the given figure $AB=BC=CD$. If $PQ$ bisects $\angle APB$, then prove that $AD||PQ$. My attempt: $$AB=BC=CD$$ $$\angle APQ=\angle BPQ$$ $$\text{arc } AB=\text{arc } BC$$ $$\angle ADB=\...
0
votes
1answer
48 views

Circle Problem:Which of the following are true

If the circle $x^2+y^2+2gx+2fy+c=0$ cuts the three circles $x^2+y^2−5=0$, $x^2+y^2−8x−6y+10=0$ and $x^2+y^2−4x+2y−2=0$ at the extremities of their diameters, then which of the following are true ? $c=...
0
votes
1answer
56 views

Bending a line segment

REVISED QUESTION With the help of the existing answers I have been able to put together this clearer animation, and I asked this question to discover the shape is called a cochleoid. What I am ...
2
votes
1answer
108 views

Connection of two objects in coordinate system

Please take a look at the following picture: I want to connect two objects by a line. This line has to start and end on an the red lines of the objects and are not allowed (at least it should be ...
1
vote
2answers
33 views

In the given figure, $X$ and $Y$ are two centres

In the given figure, $X$ and $Y$ are two centres of two circles. They touch each other externally at a point $S$. $AB$ be the common tangent of both circles. $O$ be the centre of the third circle ...
0
votes
0answers
32 views

How can I calculate the number of corners in a prism? [closed]

For any prism whose cross-section is a regular polygon of n sides, can it be shown that there is one corner for every m edges? If not, can an expression be found to calculate the corners?
8
votes
4answers
171 views

Proving $\frac{1}{\cos^2\frac{\pi}{7}}+ \frac {1}{\cos^2\frac {2\pi}{7}}+\frac {1}{\cos^2\frac {3\pi}{7}} = 24$

Someone gave me the following problem, and using a calculator I managed to find the answer: $$\frac {1}{\cos^2\frac{\pi}{7}}+ \frac{1}{\cos^2\frac{2\pi}{7}}+\frac {1}{\cos^2\frac{3\pi}{7}} = 24$$ ...
0
votes
1answer
26 views

A, B, C, D and E are five vertices of a cuboid. AE is a diagonal of the cuboid. Find the coordinates of E. [closed]

All coordinates are with respect to origin $O$. $A = (1,2,-1) B=(0,-3,-8) C = (4,-1,-10) D= (-4,2,-11)$ Find $E.$
1
vote
1answer
41 views

Connecting $N \times N$ dots with straight lines

Many of you have likely seen the Connect 9 dots with 4 lines puzzle. My problem is similar, but more generic. Given an $N\times N$ square of points, what is the minimum number of lines needed to ...
-1
votes
0answers
12 views

Identify the given venn formula and describe its shape [closed]

Here is the image of the question.
1
vote
0answers
27 views

How to scale a coordinate system?

I have a bunch of coordinates and I draw them on a Bitmap Image to then display them on an Image control in a Wpf application. Given data to take into consideration: My Bitmap File is ...
0
votes
1answer
37 views

On ruler-only construction

There is a ruler with two different labels on it (1 inch, for example). The task is to find out if it is possible to construct a perpendicular to the given line. I have found a way to construct a ...
0
votes
1answer
29 views

How to get neighbor points in high dimensional space?

I am writing to write an algorithm to do neighbor points search (so the implicit form will not work). We define neighbor points as the points with Euclidean distance $r$ to a given point. In 2D, for ...
0
votes
2answers
20 views

Evaluate the angle x in terms of angles and y and z?

I need help with this problem and problems like this. I know that to solve this problem I have to find which angles are the same and etc, but how do I know that. How can I see that two angles are the ...
0
votes
0answers
30 views

Simplification challenge (cross products)

In a calculation involving three 3D-vectors $A$, $B$, $C$, the following term appeared $$ 2\langle A\times B, C\rangle^2-\langle A\times B + B\times C + C\times A, (B\times C)\langle A, A\rangle + (C\...
13
votes
2answers
1k views

What is the name of the circle that is tangent to three mutually-tangent circles centered at the vertices of a triangle?

I want some information about the little 'tangent circle', but I don't have its name to search for it in the internet. What is it called?
1
vote
1answer
50 views

Finding the orthocentre of a trinagle.

Now, I know this has been asked here but my question is something else so please bear with me. Question:- If the vertices of a triangle are represented by $z_1, z_2, z_3$ respectively then show ...
1
vote
0answers
32 views

12 points circle associated with a cyclic hexagon

When I research this problem A chain of six circles associated with a cyclic hexagon. I found the followings result. Let $ABCDEF$ be a cyclic hexagon. Let $A_1$ be any point on $AD$, the circle $(...
0
votes
1answer
10 views

Finding an equation for a plane passing through three points (Serge Lang Example Problem)

An example problem from Serge Lang's Calculus of Several Variables (pg. 30-31): Example 3. Find the equation of the plane passing through the three points $$ P_1 = (1,2, -1), P_2 = (-1, 1, 4),...
1
vote
1answer
26 views

Is there a simple way to decide if a hyperboloid is one-sheeted or two-sheeted, given the quadric equation?

Let us say that we have a quadric equation, whose solution set lies in $\mathbb{R}^3$, and you know it's a hyperboloid. Is there a way to analytically decide through a criterion if the hyperboloid is ...
3
votes
1answer
55 views

Compute the intersection of the offsets of two tangents between two circles

Pardon the amateurish notation and what's probably a very simple puzzle for most on this site. Still, the solution eludes me.... I have 3 circles $C_R, C_G, C_B$ as shown in the diagram. I know the $...
7
votes
3answers
665 views

Where do you see cyclic quadrilaterals in real life?

I've just been studying cyclic quads in geometry at school and I'm thinking see seems pretty interesting, but where would I actually find these in the real world? They seem pretty useless to me...
6
votes
4answers
620 views

'Concentric' parabolas — two parabolas that have a constant vector distance

I am trying to 'draw' a two-dimensional path in the shape of a semi circle with thickness d in the xy-plane. The way I would like to do this is to have two parabolas, $f(x) = ax^2$ and $g(x)= bx^2 +cx ...
0
votes
1answer
68 views

Loci of intersection points of two curves

There are two continuous, negatively-sloped curves,A and B. They intersect at least once ,say at $(x,y)$. If I introduce a third curve C, whose X axis intercept has a higher magnitude than that of B, ...
3
votes
1answer
28 views

If $XY$ is diameter, $PR\perp XY$..

In the given figure, $XY$ is the diameter, $PR\perp XY$ and $PC\perp XQ$ then prove that $2AB=QR$. My Attempt $P,X,A and C$ are Concyclic points. So, $PXAC$ is a cyclic quadrilateral. $\angle ...
0
votes
2answers
14 views

Given x coordinate difference find the angular difference

A point on a circle moved horizontally by $x$. How to find $\alpha$ in the picture below, knowing $x$, circle radius and center? I'm pretty sure this is doable but I just can't reach the solution. ...
6
votes
3answers
106 views

Reflection relating two subspaces

Let $S_1, S_2 \subseteq \mathbb{R}^n$ be two linear $k$-dimensional subspaces. Does there always exist a hyperplane $H$ such that $S_1 = R_H S_2$, where $R_H$ denotes the orthogonal reflection across $...
0
votes
0answers
22 views

Clarification Needed:Equation Of Refracted Ray/Line

A ray of light is sent along the line $2x-3y=5$.After refracting across the line $x+y=1$ it enters the opposite side after turning by $15^0$ away from the line $x+y=1$.Find the equation of the ...
0
votes
1answer
55 views

Determine the varying area of the shaded region

How to estimate the area of the shaded region shown in the attached Figure? Note that in the figure, $p$ has a maximum and minimum values of $p_a$ and $p_b$ respectively. Moreover, $p$ follows a ...
1
vote
1answer
39 views

A rectangle of a given aspect ratio inscribed in a hexagon.

I'm trying to find the largest rectangle of a given aspect ratio that can be inscribed in a hexagon. I'm able to sort of walk through the problem in reverse, i.e. given an x, I can calculate the ...
4
votes
0answers
47 views

On the relationship between $\text{SL}_2(5)$ and $A_5$ [duplicate]

I have two questions. What is the quickest way to see from scratch that $\text{SL}_2(5)/\{\pm I\}$ is isomorphic to the alternating group $A_5$? Does $\text{SL}_2(5)$ have any subgroups isomorphic ...
8
votes
0answers
55 views

Finite subgroup of $\text{SO}(3)$ acts on set of points on unit sphere in $\mathbb{R}^3$ which are fixed via some nontrivial rotation in $G$

Let $G$ be a finite nontrivial subgroup of $\text{SO}(3)$. Let $X$ be the set of points on the unit sphere in $\mathbb{R}^3$ which are fixed by some nontrivial rotation in $G$. I have two questions. ...
1
vote
1answer
23 views

Find the ratio of diagonals in Trapezoid

Given $ABCD$ a rectangular trapezoid, $\angle A=90^\circ$, $AB\parallel DC$, $2AB = CD$ and $AC \perp BD$. What is the value of $AC/BD$ ? Attempts so far: I have tried using the ratio of the areas ...
0
votes
0answers
13 views

efficiency of different whole-number-mass-to-a-power in balancing a regular triangle/tetrahedron

I saw this qustion: http://puzzling.stackexchange.com/questions/186/whats-the-fewest-weights-you-need-to-balance-any-weight-from-1-to-40-pounds Suppose you want to create a set of weights so ...
3
votes
1answer
32 views

Geometric Shapes that can be placed inside itself

My questions title may need to be improved, and I am highly open for recommendations. Also if this is the incorrect community to post in, I would be happy to be directed to the correct one. I am ...