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
29 views

How to find the intersection point of two moving circles?

I'm trying to develop a simulation in C#, and I have to find the intersection (or collision) point of two moving circles, in 2D space. Actually one of my circles will be stationary, so only one of ...
5
votes
0answers
30 views

what is vector $(\vec{a}\cdot \vec{b})\vec{c} + (\vec{b}\cdot \vec{c})\vec{a} - (\vec{c} \cdot \vec{a})\vec{b}$

Suppose we have three non orthogonal vectors in $R^3$ as $\vec{a}, \vec{b}, \vec{c}$. The vector of $(\vec{b}\cdot \vec{c})\vec{a} - (\vec{c} \cdot \vec{a})\vec{b}$ is in the plane spanned by ...
0
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0answers
28 views

What is a geometric structure?

Every elementary book on abstract algebra usually begins with giving a definition of algebraic structures; generally speaking one or several functions on cartesian product of a point-set to the set. ...
2
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3answers
37 views

homotopy groups of wedge sum

Let $X_\alpha$ be connected CW-complexes. Then from Hatcher's book, $$\pi_{n}(\prod_{\alpha} X_{\alpha})=\prod_{\alpha}\pi_{n}(X_{\alpha}).$$ Is it true in general $$\pi_{n}(\bigvee_{\alpha} ...
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votes
1answer
25 views

Area Of Polygon Whose Edges Are In Given Distance From A Given Polygon Edges

I'm handling a problem which I find quite difficult to solve; My input is a changing number of coordinates (real GPS coordinates), usually I get 4-8 coordinates, and another number,which indicates a ...
0
votes
1answer
21 views

Point on a sphere - translating reference axis

I have a point on unit sphere described by two angles : zr = Angle of rotation around the z-axis zi = Angle of inclination from the z-axis The problem that I have is that the data I need to use is ...
3
votes
1answer
36 views

Tiling squares with L-Trominoes

Is there a simple proof that any square besides a 3x3 square with area divisible by 3 is tileable with L-trominos?
0
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0answers
11 views

Geometry problem - lines/piece wise defined function

I am bothered by a certain problem. Suppose $x$ lines are drawn in the plane where no two of them are parallel and no three or more meet at one point. How many unbounded regions are there? Its pretty ...
3
votes
1answer
32 views

proof involving a triangle with a point inside it. [duplicate]

Suppose we have a triangle, call it triangle $XYZ$, and a point $W$ inside triangle $XYZ$. How would I prove that $XY + YZ > XW + WZ$? So the way I labeled everything, point $X$ is the bottom left ...
2
votes
1answer
27 views

Show that this construction is a parallelogram.

Let $ABC$ be a triangle. The middle of the segment $BC$ is denoted by $M$ and the centroid of $ABC$ is rated $G$. We construct $G'$ on the line $GM$ such that $|GM|=\frac{1}{2}|GG'|$ and ...
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0answers
54 views

Can one prove existence of incommensurables without the Pythagorean theorem?

Euclid's proof that the side and the diagonal of a square have no common measure, probably going back to Pythagoreans, reduces it to proving the irrationality of $\sqrt{2}$. This reduction uses the ...
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0answers
24 views

Distance of the plane relative to the base of a Pyramid.

Consider a pyramid is cut by a plane parallel to its base. Question: What is the distance of the plane relative to the base so that the volume of the truncated pyramid so formed is $\frac{3}{8}$ of ...
-1
votes
1answer
25 views

How to find the number of revolutions? [on hold]

The wheel of a bus has a radius of 50cm. How many revolutions of the wheel will the bus need to cover a distance of 3km ?
1
vote
2answers
53 views

Why must closest approach occur when relative velocity is perpendicular to motion?

The first part i) I can solve correctly, but I need some advice and intuition on how to solve the second part ii). Here is the mark-scheme for the question: But for part ii) I do not understand ...
0
votes
1answer
40 views

Circles and tangents and circumcircles

Question: Tangents drawn from the point $P(1, 8)$ to the circle $x^2 + y^2 -6x -4y -11=0$ touch the circle at the points $A$ and $B$. What is the equation of the circumcircle of the triangle $PAB$? I ...
1
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0answers
48 views

Would this thinking about the dot product hold?

Background today I completed the chapter on the dot product of vectors. But in trying to figure out exactly what the dot product is. I came to the conclusion that it can be interpreted as the length ...
1
vote
2answers
107 views

Tricky geometry proof

If a,b,c belong to the interval $(0,1)$ and $ab + ca + bc = 1$, prove that $$\frac{a}{1-a^2} + \frac{b}{1-b^2} + \frac{c}{1-c^2}\ge\frac{3^{3/2}}{2}$$ How would you go about solving such a problem?
0
votes
1answer
23 views

Area of a triangle - straight lines

Question: What is the area of the triangle formed by the line $x + y = 3$ and angle bisectors of the pair of straight lines $x^2 - y^2 + 2y = 1$. Well I really have no idea how to even start the ...
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0answers
17 views

Is there a way to generate a regular X-faced object? [on hold]

For example a dodecahedron, but generated in 3d software with a variable amount of faces
5
votes
1answer
49 views

Shortest path between two points via two disks

Hallo everybody, I have the following problem regarding shortest paths in $R^2$. Suppose you are given two points $p$ and $q$ and two unit disks, as in the picture. I am looking for a path from ...
1
vote
2answers
42 views

bottom half of a sphere

if a sphere is of the equation $r^2 = x^2 + y ^2 +z ^2$ and we want to find a hemisphere does that just involve setting a limit to be half of the radius ? For example if we know that the centre of ...
1
vote
0answers
37 views

geometrical consequences of nonpositive or negative Ricci curvature.

well,that's pretty much the question. I'd like to know if somebody could point me out if there's any geometrical implications following an upper bound on the Ricci curvature on a riemannian manifold. ...
0
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0answers
24 views

Geometrical problem of maximum area of smallest triangle formed by 3 points in a distribution of n points on an R^2 plane

Another way of stating the gist of the question is: find the arrangement of n points such that one obtains the largest ratio between the area of the smallest triangle formed by three points to the ...
-1
votes
1answer
17 views

AB is a diameter of the circumcircle of triangle APB. [on hold]

AB is a diameter of the circumcircle of triangle APB.N is the foot of the perpendicular drawn from the point P on AB.If AB=8cm and BP=6cm,then the length of BN is
1
vote
1answer
41 views

What is a geometric shape?

I thought that the concept of "gemoetric shape" is clear enough - squares, ellipses, triangles, you know. But then I found several papers, such as this one, which define "shape" as "an ...
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votes
1answer
26 views

How to find slope on line that known only point and angle

How to find slope on line that known only point and angle Image will describe more clearly I'm wont to find the orange line slope to find point on it ( b , c , d ) suppose that A and angle are ...
0
votes
2answers
24 views

Two circles with the same radius r intersects each other and one passes to the centre of the other.

Two circles with the same radius r intersects each other and one passes to the centre of the other.Then the length of the common chord is can someone help me I think its answer is r because passes ...
2
votes
3answers
43 views

How to find the radius of this middle circle arranged as shown.

There is this maths competition geometry problem and my approach. And this is my initial approach. From the picture, the shaded circle looks slightly bigger. What we are looking for is the $x$ ...
1
vote
2answers
40 views

A geometric question about drawing lines in a plane

Suppose we were to draw lines in a plane such that no two of them are parallel and no three or more meet at one point, so in other words there is only double intersections. If we drew $x$ lines how ...
2
votes
1answer
12 views

Angles of which locations of points corresponding to distances intersect each other

Original problem : In the $XY$ plane,Let $P_1$ and $P_2$ two points with coordinates $(-1,0)$ and $(1,0)$. $C_1$ is the locus of points whose sum of distances to $P_1$ and $P_2$ is $4$. $C_2$ is the ...
2
votes
2answers
75 views

Example of a proof using the axiom of commensurability

I'm teaching our intro to proofs course (well, one of them) and one of the classic illustrations of an overturned "axiom" is the Greek axiom of commensurability, which stated in geometric terms the ...
3
votes
3answers
46 views

Right-Angled Isosceles Triangle covering puzzle

Consider a RAIT (right-angled isosceles triangle), from which we remove a RAIT smaller than half its area by a cut perpendicular to the hypotenuse, like this: How many RAITs are required to cover ...
1
vote
1answer
21 views

Algorithm for solving line line intersection in 3d

I am trying to find an algorithm that a computer can execute that finds the intersection point between two lines each defined by a point on the line and a direction vector. Does anyone know of one? It ...
3
votes
1answer
20 views

inflection point of cubic bezier with restrictions

Say you have this type of cubic Bézier curve: The 4 control points A,B,C,D have restrictions: A & B have the same Y-axis coordinate C & D have the same Y-axis coordinate B & C have ...
0
votes
1answer
29 views

Why does $n$ have to be a perfect square for me to construct an equilateral triangle out of equal smaller ones?

If I have $n$ unit squares and want to build a bigger square out of the ones I already have, it is obvious that $n$ itself has to be a perfect square. But after doing some elementary math it turned ...
2
votes
2answers
47 views

How to determine whether three ellipses have at least one common intersection point or not?

How to establish a criterion described in equation so that it is easy to determine whether three ellipses have common intersection area (point) or not? Update
16
votes
6answers
2k views

Can one deduce whether a given quantity is possible as the area of a triangle when supplied with the length of two of its sides?

Recently I have found a question like following: In triangle $ABC$, $AB=AC=2$. Which of the following could be the area of triangle $ABC$? Indicate all possible areas: [A] $0.5$ [B] $1.0$ ...
0
votes
1answer
29 views

Finding equations for plane figures using complex coordinates

I have to find conditions defining the following plane figures: Where: $a=3$ and $b=7$ I know that circumference form is: $$\left |z-z_0 \right | =b$$ So, for c. with center $(3,3)$ and radius ...
0
votes
2answers
33 views

No of triangles in a square which contains all the m points?

Given a square $A(0,0)$, $B(0,n)$, $C(n,n)$ and $D(0,n)$ in $X­Y$ plane and a set of $m$ points. The $m$ points strictly lie inside the square $ABCD$. It is clear that there are $4n$ integer points on ...
1
vote
1answer
14 views

Avg # of Rectangle Intersections in 2D Field

So imagine I have a large 2D field. Thousands of small rectangles overlay the field. The field is much larger than the rectangles. The rectangles are placed randomly in the field such that they may or ...
2
votes
1answer
35 views

At the instant of release of an object from rest. Is the only force that can act its weight? [on hold]

This is the third question from a mechanics exam past paper: I can do parts i) and ii) but for iii) in finding the angular acceleration, i used $C=I\alpha$, where $C$ is the applied couple or ...
0
votes
0answers
8 views

Extract equations of dependency between two projected views

Regarding question Finding equation of an ellipsoid, the answer says that we have the following equation between projections on XY & XZ plane: $$\frac{Z_3^2}{Z_2} - Z_1 = \frac{Y_2^2}{Y_3} - Y_1$$ ...
0
votes
2answers
31 views

Quadrilateral angles when inscribed inside square

"If we can draw a quadrilateral inscribed in a circle, its opposite sides must sum to 180∘." Why is this?
0
votes
2answers
41 views

Calculate the unknown coordinates of a point $B (x_2,y_2)$ on a line with given distance from a known point $A(x_1,y_1)$

I have a line which represents a cross section. I have the coordinates of on its starting point. I need the coordinates of the end point of that cross section line. The distance between these two ...
0
votes
2answers
40 views

Why does resolving forces in one direction give a completely different answer to resolving the opposite way?

I can solve parts i), ii) and am able to show that $R=0$ for part iii). In this question $g$ is the acceleration of free fall taken to be $9.8$ Using Newtons 2nd law [$F=ma$] for the last part I ...
2
votes
1answer
17 views

Does simply-connected imply measureable?

The famous examples of non-connected sets involve a sophisticated selections of points from a ball (or another object). This raises the following question: if a certain object in a Euclidean space is ...
1
vote
1answer
21 views

Prove that line segments are parallel.

Prove using slope of lines that line segment joining the midpoint of $\overline { AB}$ and $\overline{AC}$ in $\Delta ABC$ is parallel to $\overline {BC.}$ Need to prove using slope of lines means I ...
3
votes
3answers
48 views

$T^2\times S^n$ is parallelizable

This is taken from a UCLA Geometry/Topology qualifying exam. How would one prove that $T^2\times S^n$ is parallelizable for all $n\geq 1$? Is there a way to find $n+2$ linearly independent vector ...
1
vote
1answer
45 views

Geometry question pertaining to a plane going through the skeleton of a cube

My question is as follows: a plane that has taken the shape of a pentagon is intersecting the skeleton of the cube. Or I guess we could think of it as a cross section. Points $M$ and $N$ were used ...
2
votes
0answers
39 views

Find circles that completely cover a polygon minimizing the amount of space covered outside the polygon

I have an arbitrary polygon that I need to roughly represent using circles. Any point inside the polygon must lie inside a circle. There will be points outside the polygon that will fall under a ...