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Questions tagged [geometry]

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

5
votes
2answers
14 views

Overlapping circles covering polygon

While working in GeoGebra I noticed something odd. I had a triangle with a point inside and the point was connected to each of the vertices. For each vertice I had drawn the circle passing through the ...
1
vote
1answer
15 views

Distance between a point and low-dimensional sphere

Is there a way to analytically calculate the distance between an arbitrary point $\mathbf{x}\in\mathbb{R}^n$ and a low-dimensional sphere embedded in $\mathbb{R}^n$, say one aligned with the axis ? ...
1
vote
2answers
87 views

$\mathbb CP^1 \approx S^2$ proof check

I wanted to give a whole proof of this fact as I was not able to find a detailed one myself. I have the feeling that such a proof has been asked quite frequently by several users and I hope this may ...
0
votes
1answer
40 views

How many points are needed to define a circumference?

This doubt comes from a combinatorics problem in a textbook, which states: Consider two strictly parallel lines and seven dots, four of which are over one of them, and three over the other. Three ...
0
votes
1answer
22 views

How would be a formal answer for an automata geometry problem?

Let an Automata A sitting on a point O (0,0) and turned to North. That Automata can execute only any combination of three different commands in each step: Move one unit forward Turn 90 degrees ...
2
votes
0answers
34 views

Are there always two circles that together surround or intersect all points in the following scenario?

Consider $N$ points in $\mathbb{R}^2$ and $\binom{N}{2}$ circles, one for each pair of points such that it intersects both. Is it always possible to pick two of these circles that together surround or ...
14
votes
2answers
617 views

Magnifying glass in hyperbolic space

My grandmother used to read with a magnifying glass. What (an ideal) magnifying glass does, is basically a homothety: it scales the picture by some factor. Now, in a hyperbolic space there is no such ...
1
vote
2answers
714 views

Rectangle inscribed in a circular sector of angle 60

My apologies if this has been asked before. Given a circular sector, say of radius $r$, with internal angle $60^{\circ}$, construct a rectangle inscribed in that sector so that the length of the ...
0
votes
1answer
18 views

How to show this property of Mobius Transformations

How can I show the following property of complex FLTs? $$ F_{XY} = F_X ◦ F_Y$$ where, $X, Y ∈ SL_2(\Bbb R)$ I know the inverse map of $F_X$ exists. This map is also a continuous map of $\Bbb ...
0
votes
1answer
15 views

Plane cutting a pyramid

Pyramid with equilateral triangle as a base, length of side of pyramid is $s=3$(not a base side). Plane goes through pyramid, and contains base edge, and is normal to a side of pyramid. If surface ...
0
votes
0answers
37 views

Changing rotation center

Things that we have: 2 dimensions, a object with it's coordinates (object P1), it's rotation center (pivot) C1. After that lets rotate it at pivot C1 by known angle A. Now let's move that pivot by ...
5
votes
3answers
163 views

Side length of a quadrilateral incribed on a circle

I've been doing math for 10 years now, yet every so often I get stumped by a "basic" high school question. This is one of those times. Here's the question: Part a is easy; we apply the cosine rule ...
1
vote
1answer
26 views

Find the parabola given two points and $y$-max (no axis of symmetry)

Given $(0,0)$ origin, point $(3,2)$ and $y\text{-max} = 5$, find the parabola. I tried to shift point $(3,2)$ down to $(3,0)$ so that it can become symmetric to origin. Then the vertex would be $(5,\...
0
votes
2answers
31 views

Pair of lines problem

If the pair of straight lines $x^2+2xy+ay^2$ & $ax^2+2xy+y^2$ have exactly one line in common, then the combined equation of the two lines is given by A. $3x^2+8xy-3y^2$ B. $3x^2+10xy+3y^2$ C. ...
-2
votes
0answers
13 views

How to calculate vertex, focus, axis etc. from such type of ellipse equation 3x²+8y²+12xy-18x-32y+23=0? [on hold]

Is there any way to find vertex , focus, axis, centre etc from this type of non ideal ellipse, hyperbola or parabola?
3
votes
1answer
961 views

Distances to line passing through the centroid of triangle

Let $p$ be a line that pass through the centroid of a triangle $ABC$. Unless the line pass through one vertex, then $2$ verices are one side of the line, while the third one is on the other side. ...
0
votes
1answer
45 views

Finding the lengths of $AC$ and $AD$

Triangle $ABC$ is right-angled at $A$. The angle bisector from $A$ meets $BC$ at $D$. If $CD=1$ and $BD=AD+1$, find the lengths of $AC$ and $AD$. I have tried to set up a equation with $AD$ and $AC$ ...
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votes
0answers
22 views

A problem on combinatorical geometry . [on hold]

Hey could someone help with this exercice, i have tried everything but nothing seems to suite work. Any help would br appreciated : Let ABCDEF be a convex 6 sided polygon of sides 1, prove that at ...
-1
votes
0answers
39 views

Geometry, does this shape have a name?

A Sphere with Diameter 1 perfectly inscribed in A cube with sides of 1, Removing the sphere and splitting the cube on the faces we then have 8 identical "Corners" with three sides being a tetrahedron ...
0
votes
2answers
41 views

What is the Geometric meaning of vector norm in Rn n>3

My question is related to the length of the vector , Sorry it may seem stupid for you as i come from engineering background not mathematics background For Vectors up to 3 dimensions (can be ...
1
vote
1answer
42 views

$|AB|+|BC|=l$ , find the position of $A,B,C$ maximize area of quadrilateral $OABC$ ? [on hold]

The given condition is $|AB|+|BC|=l$ In a 2D coordinate system, point $O(0,0)$ is origin. With the following points: $A(0,a),a\geq0$ $B(b_1,b_2), b_1\geq0,b_2\geq0$ $C(c,0), c\geq0$ and the ...
0
votes
0answers
10 views

Chord of contact in polar coordinate

Can you please help me to derive the equation of chord of contact for a circle in polar coordinate? I have found the equation in cartesian coordinate but I cannot map that into the polar coordinate. ...
0
votes
3answers
476 views

Geometry proof problem (high school)

I have an upcoming chapter test and this was one of the practice problems. Can someone guide me? Given: Isosceles $\triangle ABC$ with $AB$ congruent to $AC$; $AD$ is not a median of $\triangle ABC$...
0
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0answers
18 views

Möbius Transformation on Riemann Sphere

Just started learning about Möbius Transformations myself, and I wanted to know what kind of transformation on the Riemann Sphere would preserve $S^1$ (unit circle) as a set? What would the conditions ...
0
votes
1answer
24 views

Showing complex FLT to be continous

If I have the complex Mobius Transformation, $$ F_X (z) = \frac {az + b}{cz + d} $$ how can I show that the transformation is continous w.r.t the metric on $\Bbb C \cup \infty $ which is ...
0
votes
1answer
40 views

If P(Q)R and Q(R)S, prove P(Q)S

$$\text{If }\ P(Q)R\ \text{ and }\ Q(R)S,\; prove\ P(Q)S$$ Axioms: (i) If P(Q)R, then R(Q)P (ii) If P(Q)R, then $\sim$(P(R)Q) and P $\neq$ Q (iii) There are at least two distinct ...
0
votes
0answers
8 views

Affix of a point [on hold]

Let $A(a)$, $B(b)$ and $E(1)$ three points of the unit circle $\mathbb{U}$. Let $P(p)$ the point of $(AB)$ so that $(AB)\perp (EP)$. Find the affix $p$ of the point P.
0
votes
1answer
12 views

Stereographic Projection question

I want to check if stereographic projection formula, $$\Omega (x,y,z) = (\frac {x}{1-z}) + (\frac {y}{1-z})i$$ matches the following description: We can identify the complex plane with a ...
0
votes
1answer
676 views

Finding the area of a triangle when only a the shaded part and two sides are known

I tried finding the height of the shaded triangle, which I calculated to be 5. Then I tried solving for the area of the non-shaded triangle, and I got 10 as my final answer. Am I correct?
0
votes
1answer
30 views

Isometric Drawing Tool: Converting 2D information to 3D

I was drawing with the NCTM Isometric Drawing Tool, and produced the image seen here. I also noticed that it is possible to view the isometric drawing in 2D, and was wondering if/how it is possible to ...
1
vote
3answers
67 views

Area of the intersection of two triangles.

Let $\triangle{ABC}$ be a triangle with $AB=5$, $BC=7$, and $CA=4$. Define $D$, $E$, and $F$, to be the midpoints of $AB$, $BC$, and $CA$ respectively. Let $G$ the intersection of the medians of $\...
1
vote
3answers
753 views

Perfect cuboid cube

Is there any proof that there is no cubic perfect cuboid? Here is a description of the problem: . I'm currently using trying to get an empty set to solve it... [ A "perfect cuboid" is one whose edges,...
2
votes
1answer
98 views
+50

Special proof for Pick's Theorem?

This is kind of a two-part question and I think one should lead to the other- Could someone show or point me to a proof of the theorem for primitive triangles. Also, let's say we have a lattice ...
1
vote
0answers
23 views

Centroid of an irregular pentagon with variable sides

I would very much appreciate any help on this important question to my work. Suppose an irregular pentagon is formed by the combination of 5 variable parameters, as illustrated in the Figure. I know ...
1
vote
2answers
41 views

How to solve an equation with truncated unknowns?

In order to provide some context, here's the problem that took me to the question: I'm attempting to fill a rectangle of known dimensions (w: width, h: height) with squares whose side is s. I also ...
-1
votes
1answer
51 views

Prove that $EF'$ passes through the circumcenter of $\triangle E'M'N'$.

$F$ is a point outside $\square ABCD$ such that $\widehat{CFD} = 135^\circ$. $FF' \perp CD$ at $F'$. $AC \cap BD = \{E\}$. $AF \cap BD = \{M\}, AF \cap CD = \{M'\}$ and $BF \cap AC = {N}, BF \cap DC = ...
0
votes
1answer
43 views

The coordinate of a circle [closed]

Suppose two circles intersect and form three regions A, B, and C. The center of circle A is (2,2) and the center of circle B is (x,y). The three regions formed by the two circles are equal in area. ...
3
votes
3answers
692 views

How many of these lines lie entirely in the interior of the original cube? [on hold]

A portion of a wooden cube is sawed off at each vertex so that a small equilateral triangle is formed at each corner with vertices on the edges of the cube. The $24$ vertices of the new object are all ...
2
votes
0answers
50 views

Cutting a Solid Torus with $n$ Planes

Question: How many pieces a solid torus be cut into with three (affine) planar cuts? A google search will quickly reveal that the answer is thirteen, as can be read about here. The picture below ...
1
vote
1answer
84 views

Finding Angles of Delta of Helicoid

Consider the helicoid $S$ given by the parametrization $$x(u,v)=(v\cos u,v\sin u,u).$$ a) Let $T$ be the curvilinear triangle on $S$ which is the image under $x$ of the triangle $\{(u,v): 0 \leq ...
3
votes
2answers
887 views

Shift in origin for polar coordinates

Consider a set of polar coordinates, $(r, \theta)$ for a plane. Let's say there is a point, at $\left(1,\frac{\pi}{4}\right)$ for which I want to define a translated and rotated coordinate system, $(R,...
0
votes
1answer
54 views

Maximal area of equilateral triangle inside rectangle.

If the perimeter of the rectangle is P, what would be the maximal area of the equilateral triangle if: - One of the sides of the triangle coincides with one of the sides of the rectangle - We remove ...
35
votes
8answers
3k views

Show that a generalized knight can return to its original position only after an even number of moves

Source: German Mathematical Olympiad Problem: On an arbitrarily large chessboard, a generalized knight moves by jumping p squares in one direction and q squares in a perpendicular direction, p, q >...
0
votes
1answer
25 views

General equation for projection of regular grid onto a line?

I have a regular grid of points in $xy$, say a square grid, and I want to make an orthonogal projection onto a line through the origin, with slope $\tan \alpha$: I would like to derive a mathematical ...
0
votes
1answer
16 views

Finding Overlap of polygons in 3D space

I'm trying to find the amount of "overlap" between two (or more) polygons in a 3D space. The planes all have vector normals pointing in the same direction, so they are guaranteed to be parallel to ...
2
votes
1answer
113 views

Exact Differential Equation Geometry

In a variety of contexts, I have noticed hints of a strong connection between exact differential equations and machinery from multivariable calculus. From another question, I have gathered that the ...
0
votes
0answers
20 views

Decipephering Notation and plugging in values for ellipse formula

Premise I had asked a question on stats exchange about calculating error ellipses for a given scatter plot. I got an answer that seems acceptable, but I'm having trouble implementing it because my ...
0
votes
1answer
44 views

Is a triangle a conic section? If not, why? [on hold]

If I take the intersection for a=180° the section described is a triangle, but a triangle is never defined as a type of conic section.
0
votes
1answer
40 views

Pythagoras theorem in oblique coordinates

consider two oblique oblique basis vectors of unit length $\vec{r_1}, \vec{r_2}$ then any vector $\vec{v} = p\vec{r_1}+q\vec{r_2}$ define the dot product between two vectors a and b as $|b|$ (ie ...