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

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0
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2answers
11 views

Find a point on a line that is also the third vertex of a triangle

I am interested in finding the $(x, y)$ coordinates for the point, $C$ in the figure below, which is also on the line showing going through the points, $B$ and $C$. I believe this problem has a unique ...
2
votes
1answer
26 views

Relationship between Surface Area and Volume

Question: Is there a general relationship between surface area and volume analogous to the below examples? Example 1. Consider a ball $B$ centered at the origin of a spherical coordinate system. The ...
0
votes
1answer
19 views

What is the length of the line joining the mid–points of PQ and RS for this given trapezium $PQRS$?

$PQRS$ is a trapezium, with $PQ$ parallel to $RS$. $PQ = 20$ cm, $RS = 3$ cm, $PQR = 300$ cm and $QPS = 600$ cm. What is the length of the line joining the mid–points of $PQ$ and $RS$?
1
vote
1answer
210 views

Prove that lines intersecting parallel similar triangles are concurrent

Suppose $\triangle ABC$ and $\triangle A'B'C'$ are two similar but non congruent triangles such that $AB$ is parallel to $A'B'$, $AC$ is parallel to $A'C'$, and $BC$ is parallel to $B'C'$. Prove ...
-2
votes
1answer
35 views

Lines through vertices of regular hexagon and regular pentagon [on hold]

ABCDEF is a regular hexagon and ABPQR is a regular pentagon of equal sides that are joined with each other through one common side AB. If the lines from neighbouring points of A (i.e. from F and R) ...
1
vote
0answers
17 views

Parametrization of a surface

I am given the curve $ a (u) = (cos(u), sin(u), u) $. I am asked to write the parametrization of the surface obtained intersecting this curve with lines orthogonal to the z axis. How to do this? ...
0
votes
2answers
24 views

Hypotenuse and angle ratio relationship

In triangle ABC $\angle BAC=90$, $\angle ABC$:$\angle ACB $=1:2 and AC = 4cm. Calculate the length of BC. I tried this by constructing an equilateral triangle as in the figure. I am interested in ...
29
votes
2answers
747 views

Geometry problem involving infinite number of circles

What is the sum of the areas of the grey circles? I have not made any progress so far.
0
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3answers
32 views

Finding the variable of a coordinate point on a circle

This might be a very simple question but I am having trouble figuring it out, so if anyone can explain: A circle is marked with three points A(-3,2),...
6
votes
0answers
81 views
+50

Which power means are constructible?

The three classic Pythagorean means $A$, $G$, $H$ (arithmetic, geometric, and harmonic mean respectively) of positive real $a$ and $b$ have a cute geometric construction, as does the quadratic mean ...
0
votes
1answer
223 views

How to show that a line pass through a point?

How to show that a line pass through a point? Two fixed straight line $OX$ and $OY$ are cut by a variable line at the points $A$ and $B$ respectively and $P$ and $Q$ are the feet of the ...
2
votes
4answers
53 views

Enclosing land by fence pieces

You have one 44-meter piece of fence and 48 one-meter pieces of fence. Those fences are straight and cannot be bent. What is the biggest area you can enclose with those fences on a two dimensional ...
0
votes
1answer
26 views

Solid Angle Integration

Can somebody explain the equivalence between integrating over the surface of a unit sphere and integrating over solid angle? I have been trying to understand the following transformation using a ...
1
vote
1answer
675 views

Calculating points in an arc

Hi I'm trying to figure out how to calculate the coordinates of a dot at a certain percentage point on an arc. Let's say the dot starts at (800, 300), the half-way points is (400, 0) and the end point ...
0
votes
1answer
23 views

Defining rotation without using angles, but as geometric transformations?

According to this article on angles, we can define rotation without using angles, and then use rotation to define angles. The relevant paragraph is at the very end: But what is a rotation? Is it ...
0
votes
1answer
58 views

Determine if one point lies between two other points on a sphere

My question is rather simple. Can I use the dot product to determine if a coordinate lies between two others? With coordinates I mean a Point P(latitude, longitude) on the surface of the sphere. I ...
3
votes
2answers
16 views

Radius of circumference tangent to square and circular sector

I would like to find the radius of the circumference shown in figure, knowing the side of the square is 5. I have decided to note said radius $r$ and the tiny diagonal bit not included in any circle ...
1
vote
2answers
21 views

Finding the points where a circle intersects an axis

A circle has the equation: x²+y²+4x-2y-11 = 0 What would be the coordinates of the points where the circle intersects with the y-axis and how would you calculate it?
0
votes
1answer
30 views
+50

rotate secondary Vanishing points to the primary vanishing points to find new length of object

all though only the 2D data is available, the best way to think of this problem is a piece of paper pinned at one corner to a wall, but the paper is sitting at an angle to the wall, see illustration ...
4
votes
3answers
2k views

What is the precise definition of a rigid shape?

Wikipedia's section on rigid shapes does not appear to actually define what a rigid shape is. Rather it defines 'same shape' and 'rigid transformations' without giving any definitions of what is ...
0
votes
0answers
12 views

Interior and exterior of a polygon in Hilbert axioms

First of all sorry for my bad English. Correct me if needed. I can't prove one theorem from Hilbert's "Foundations of Geometry". Here is the quote: Theorem 6. Every simple polygon, whose vertices ...
0
votes
3answers
31 views

How to find a 4D vector perpendicular to 3 other 4D vectors?

In 3 dimensions it is possible to find a vector c (one of infinitely many) perpendicular to two vectors a and b using the cross product. Is there any way of extending this to 4 dimensions, i.e. given ...
1
vote
2answers
144 views

Is there any reason why $4-\pi$ is quite close to $\frac{\sqrt{3}}{2}$?

In this question obviously the error of our "approximation" is $4-\pi=0.858...$ . I tried to reconstruct the false argument with $\tau=2\pi$, and the error in that case would be $8-\tau=1.716...$, ...
2
votes
4answers
108 views

Product of reflections is a rotation, by elementary vector methods

Let $\mathbf{u}$ and $\mathbf{v}$ be two 3D unit vectors. The transform that performs reflection in the plane normal to $\mathbf{u}$ is given by $$ T_{\mathbf{u}}(\mathbf{x}) = \mathbf{x} - ...
0
votes
2answers
285 views

Maximum volume of parallelepiped

Find the dimensions of the parallelepiped of maximum volume circumscribed by a sphere of radius R. I would normally be familiar with this using lagrange multipliers, but how do I do this? It ...
0
votes
2answers
59 views

Pentagon with two right angles (aka Van Aubel's Theorem)

My problem is the following: given that $ABCDE$ is a convex pentagon such that $AB=BC$, $CD=DE$, $M$ is middle point of side $EA$ and the angles $\widehat{ABC}=\widehat{CDE}=90°$, find the measure of ...
0
votes
0answers
16 views

Every polyhedron $P \ne \mathbb{K}^n$ equals an intersection of finitely many half spaces.

Currently, I am reading some lecture notes on linear optimisation. I cannot see why the following (seemingly trivial) proposition holds. (How could I understand/proove it?) Every polyhedron $P \ne ...
-2
votes
1answer
51 views

Polar Coordinate usind De Moivre’s Theorem

I need the solutions for the following problem: Find all solutions over $\mathbb{C}$ to the equation $x^3=i^2$. I tried using De Moivre’s Theorem can't get around it. Note: The question originally ...
1
vote
0answers
15 views

Cartesian to geodetic conversion of 3D bounding box - How to calculate latitude and longitude from an axis aligned bounding box

I have a geometry with its vertices in cartesian coordinates. These cartesian coordinates are the ECEF(Earth centred earth fixed) coordinates. This geometry is actually present on an ellipsoidal model ...
4
votes
5answers
512 views

Interesting geometry problem (square and two circles)

What's the area of the main square? (I think the attached picture defines the problem clearly.)
1
vote
1answer
33 views

Calculate PQ if AC = 20

I need to calculate PQ knowing that AC = 20. This is what I got so far: If I call the point between P and A, "M" and If I call the angle: $$\measuredangle{QPB} = y$$ Then: ...
0
votes
1answer
16 views

$P$ is a point on a hyperbola whose focal points are $F_1$ and $F_2$. $Q$ on the line that bisects $\angle F_1PF_2$. Prove $|PF_1-PF_2|>|QF_1-QF_2|$.

$\require{cancel}$ Sorry for the grammatical mistake in the title; it was needed to keep the title under 150 characters. $P$ is a point on a hyperbola whose focal points are $F_1$ and $F_2$. $Q$ is ...
7
votes
1answer
69 views

Direct formula for area of a triangle formed by three lines, given their equations in the cartesian plane.

I read this formula in some book but it didn't provide a proof so I thought someone on this website could figure it out. What it says is: If we consider 3 non-concurrent, non parallel lines ...
0
votes
2answers
20 views

Find the geometric locus of vertices A of the triangles ABC with the given base BC and such that $\widehat{B} > \widehat{C}$

When I tried it, I figure that the right triangle, with angle A being 90, to satisfy the question. I just don't think is quite correct. Any suggestions?
3
votes
2answers
129 views

Transforming $2D$ coordinates

Lets say from coordinate system 1, we have 3 points which consists of a triangle. The vertices are located at $(50,120) , (70,150) , (100,100)$. Now, coordinate system 2 consists also of a triangle, ...
2
votes
1answer
40 views

Ellipse like on sphere

Find the locus of all points on a sphere such that the sum of geodesic distances from two fixed points F1 and F2 on it is a constant, less than its diameter. ( When radius of sphere goes to infinity, ...
2
votes
1answer
47 views

Is it possible to accurately calculate an irregularly shaped frustum's volume?

I have the following water basin Now imagine this basin is filled with water to the top, is there anyway to accurately calculate the volume of water stored in it using only top and bottom areas A1 ...
1
vote
3answers
5k views

How do I Find all Angles of 4-sided polygon given side lengths?

I have a program that lets users draw custom 4-sided shapes using java 2d. I want to calculate the angles inside the shapes so I can rotate text to the proper angle and label each side. I am trying ...
43
votes
4answers
5k views

How far can one see over the ocean?

Since Earth is a sphere, one has only a limited visibility radius. How far is that, actually? This Q&A was inspired by this question, about whether or not Legolas can see the 24km distant Riders ...
0
votes
2answers
276 views

How to enlarge a circle?

if you are given a circle with equation $(x-a)^2 + (y-b^2) = r^2$ and it is enlarged by a factor of $3$ what would the new equation be? Would you put $2x$ an $2y$ in the place of $y$?
15
votes
2answers
395 views

Do there exist an infinite number of 'rational' points in the equilateral triangle $ABC$?

Let's call a point $P$ which satisfies the following condition 'a rational point'. Condition: Each distance $PA, PB, PC$ from a point $P$ to three vertices $A, B, C$ of an equilateral triangle $ABC$ ...
1
vote
2answers
642 views

How do I map the torus to a plane?

Please see my answer on Perlin noise first. A bit of background. Imagine a solid texture, like an actual block of sky and cloud. If you "cut a sheet" of sky and display it as an image, you'd get ...
2
votes
0answers
41 views

Realisations of associahedra

I seem to have lost the reference to a realisation I am interested in. Hopefully someone can steer me to a paper that fully explains the realisation. For the case $K_2$(the 5-gon) the following ...
2
votes
4answers
55 views

Find the center of a circle on the x-axis with only two points, no radius/angle given

Find the center $C$ on the x-axis of the circle containing $(15,-2)$ and $(7,10)$ I can't seem to find a formula to help me solve this problem without needing the radius or the angle between the the ...
0
votes
1answer
24 views

Euclidean and rectilinear distance and nonlinearity

Can some one please explain why Euclidean distance and rectilinear distance make a problem nonlinear? Thanks
1
vote
1answer
22 views

Transformation matrix from a translated-rotated coordinate system to the general coordinate system

In Figure 1, suppose $XYZ$ (in black) as my general coordinate system and $X'Y'Z'$ (orange) as another system with parallel axes respect to $XYZ$. Consider $xyz$ (green) is my 3rd coordinate system ...
2
votes
1answer
44 views

Application of Bessel Function

I have read number of books and online literature on Bessel function. Theoretically, I have known about Bessel function. What is practical significance of Bessel function? How can Bessel function ...
-2
votes
0answers
58 views

Euclidean Geometry [on hold]

Let $ABC$ and $A'B'C'$ be two non-congruent triangles whose sides are respectively parallel . Then prove that $AA',\, BB', \, CC'$ (extended) are concurrent. Look I came across this problem in a ...
2
votes
1answer
23 views

Is it possible to create a pop-up figure that yields a truncated cone?

Problem I have a geometrical problem. Consider the cone in the figure below. Is it possible to create a two-dimensional shape that extends to the three dimensional truncated cone? The idea is to fold ...
2
votes
2answers
32 views

Number of non zero integer values of $k$ for which the points ($k,k^2)$ lies inside the triangle formed by the given three lines

Problem : Number of non zero integer values of $k$ for which the points ($k,k^2)$ lies inside the triangle formed by the lines $11x+6y+14=0$, $9x+y-12=0$, $2x+5y-17=0$ (a) $0$ (b) $2$ (c) $3$ ...