Questions tagged [plane-geometry]

Plane geometry is a subfield of Euclidean geometry, restricted to the flat two-dimensional space. Plane geometry studies shapes, ratios and relative locations of 2D figures which can be embedded into 2D plane.

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For every pair of points $x,y\in S\subset\mathbb{R}^2,\exists z\in S$ s.t. $z$ lies on the straight line formed by $x,y.$ Can $S$ be finite?

Actual question: If $S\subset\mathbb{R}^2$ contains at least $3$ noncollinear points and for every pair of points $x_1,x_2$ of $S$, there exists a third point $x_3$ which lies on the (extended) ...
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Every noncollinear $3$ points of $S\subset\mathbb{R}^2$ contains inside of it another point of $S$. Is $S$ is a dense subset of some convex hull?

Proposition: Suppose $S \subset \mathbb{R}^2$ contains at least $3$ noncollinear points. Suppose further that for every noncollinear $3$ points of $S$, the triangle formed by those $3$ points contains ...
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Are these simple statements about polygons true?

Let $P$ be an $n$-gon with $n \gt 3$. I'm looking for proofs or counterexamples for the following statements: there exist consecutive vertices $A,B,C$ of $P$ such that $\triangle ABC \cap \partial P =...
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Is this correct? Searching for the distance between two planes…

To find out the distance between the plane $E_1: 2x+5y+4z=24$ and the plane $E_2:x+5/2 y+2z=12$ we first have to look whether they’re parallel or not, we only can find out the distance if they are ...
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Five points on a constant-width curve

Is there a curve of constant width $1$ on which it is impossible to arrange the five points $A, B, C, D, E$ so that $\max(AB, BC, CD, DE, EA) \leqslant \sin (\frac{\pi}{5})$? For example, on a circle, ...
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Compact Zero-Dimensional, Planar Sets are Contained in Planar Cantor Sets?

It's known that a zero-dimensional, compact metric space $X$ embeds in the Cantor Set: Take a countable, clopen basis $(U_j)$ indexed by $\mathbb{N}$ and construct a map $f: X \rightarrow \lbrace 0, 1 ...
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Trace of $3x^2+6y^2=1$

Let $𝑆$ be the surface defined by the equation $3𝑦^2+6𝑧^2=1$. Determine the range of values of $x$ for which the $x$-traces (cross-sections) are non-empty. That is, find all $𝑎$ such that the ...
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Find tangent plane for mean squared error loss function

I would really appreciate your help on a rather simple issue that I just can't solve on my own. I'd like to visualize gradient descent for a simple linear regression problem with two parameters $\...
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Prove that $AG+GB+GH+DH+HE\ge CF.$

In the diagram, $ABCDEF$ is a hexagon with $AB=BC=CD$ and $DE=EF=FA$. Angles $BCD$ and $EFA$ both equal $60°$. $G$ and $H$ are two points taken from inside the hexagon such that angles $AGB$ and $DHE$ ...
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I would like to ask about 3R Planes. [closed]

[Hi, I have a question in my homework and I attached my solution. By any chance somebody check is it a good solution and correct answer? Thank you.][1] [1]: https://i.stack.imgur.com/XcHWC.jpg![enter ...
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Given a plane, is there a vector that points towards the highest dz when moving by dx and dy? What is that vector called?

The question came to my mind when trying to explain snowboarding up a ramp. When you don't use an edge and the snowboard is flat, it is only stable if your momentum is going straight up the ramp. You ...
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How do I find the value of k to make the equations of a line and a plane be parallel?

Given the equation of the line $r$: $r:\frac{x-2}{k}=\frac{y-1}{2}=z$ And the equation of the plane $\alpha$: $\alpha:3x-ky-z-2=0$ How do I determine $k$ such that the line $r$ and the plane $\alpha$ ...
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Is it true that the intersection of a plane $\mathbf{\pi}=(\mathbf{v}^T,k)^T$ with the plane at infinity is just a line represented by $\mathbf{v}$?

Is it true in $\mathbb{P}^3$, that the intersection of a plane $\mathbf{\pi}=(\mathbf{v}^T,k)^T$ with the plane at infinity $\mathbf{\pi_\infty}$ is just a line represented by $\mathbf{v}$? In the ...
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Rotations and paths in the plane - involves some trigonometry

Anna starts at the origin facing the positive x-axis. She can (A) move one unit in the direction she is facing or (B) stay in place while rotating counterclockwise by angle θ. At some point in a ...
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map plane to sphere

I have a programmed a plane. It is programmed so that when I walk off on edge, it wraps around to the other edge - left to right, top to bottom - making it a torus (ie "if x > width then x = 0&...
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How to rotate any plane in 3d to (XY) plane?

I want to rotate a given plane in 3d to (XY) so i can work like in normal 2d. Because if they ask like what is the equation of a circle on a given plane, i'll just rotate it to (XY) plane and apply ...
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Can Two Compact, Non-Separating Subsets of the Plane be Separated by Jordan Curves?

Suppose $X, Y \subset \mathbb{R}^2$ are compact, mutually disjoint, and neither separates the plane. How does one prove that there are disjoint Jordan curves $J_1, J_2$ such that $X$ is in the ...
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Why is a line $\mathbf{l}$ in $\mathbb{P}^2$ defined as a linear combination of its 2D null-space?

Why is a line $\mathbf{l}$ in $\mathbb{P}^2$ defined as a linear combination of its 2D null-space as $\mathbf{x} = \mu \mathbf{a} + \lambda \mathbf{b}$ where $\mathbf{l}^T\mathbf{a} = \mathbf{l}^T\...
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Triple quotient product of a triangle in the Nine Point Circle

Show that the 9-point Circle of a triangle satisfies the cyclic ordered product of segments $$ \dfrac{BP}{PC}\times \dfrac{CQ} {QA}\times \dfrac{AR}{RB}=1 $$ Show that it could be included in a ...
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How to prove that the intersection of two planes is a line (using parameteric equations)?

I'm reading the explanation to why the intersection between two planes is a line in the textbook. This seems reasonable enough, but I don't understand the last part of the proof. This indicates that ...
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Perpendicular line to a plane ( cube )

The Question I'm trying to show that $A'C$ perpendicular on plane ($C'BD$). Found that $AB = 6$ and that $A'C'BD$ and $CC'BD$ are both a tetrahedron with the same base ( triangle $C'BD$). Still I can'...
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If $K, K' \subset \mathbb{S}^2$ are compact and $K \simeq K'$, is $\mathbb{S}^2 \setminus K \simeq \mathbb{S}^2 \setminus K'$?

One of the most immediate consequences of the Alexander-Pontryagin Duality Theorem is that if $X$ is compact and $f, g: X \rightarrow \mathbb{R}^n$ are two embeddings, then $\mathbb{R}^n \setminus f(X)...
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Parallel planes: lack of intuiton

I am trying to build an intuition to understand SVMs. If we have a binary set of data which is linearly separable then we want to maximize the distance between the 2 planes: $$\mathbf{wx} - b =1$$ $$\...
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Accessible Proof of Denjoy-Riesz Theorem?

The Denjoy-Riesz Theorem states that any compact zero-dimensional subset of the plane is contained in the image of an embedded copy of the arc $[0,1]$. The only place I've seen this proved is in ...
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Prove that $ABCD$ is a trapezoid

$ABCD$ is a cyclic quadrilateral inscribed of circle $O$. The tangent line of circle $O$ at $B$ intercepts $CD$ at $K$. The tangent line of circle $O$ at $C$ intercepts $AB$ at $M$. if $AB = BM$ and $...
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Decomposition for a shear transform $(x,y) \mapsto (x+y, y)$

In theorem 15.14 of Transformation Geometry — An Introduction to Symmetry, to prove that an affine transformation is a product of strains, George E. Martin takes a shear mapping $$\begin{cases}x'=x+y \...
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Moment of Inertia of a Lamina Around the Center of Mass

I have a two-dimensional lamina in the $xy$-plane, and I need to calculate the moment of inertia around the center of mass. I know that the moment of inertia around the $x$-axis is $I_x = \int\int y^2 ...
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Set of points that satisfy the sum of the distances between a point P and points A(-2,0) and B(2,0) = 7

So, I have to find the set of points whose distance to point $A(-2, 0)$ plus the distance to point $B(2, 0)$ is equal to $7$. I am solving it this way: $ d(P, A) + d(P, B) = 7 \Leftrightarrow $ $ \...
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How to minimize exposed surface for half a pie?

I bought a large pie (of radius $R$). I cut off a half and gave it to my friend. This exposed an area or $2Rh$ -- where $h$ is the pie's thickness -- to air. I watched one Numberphile video too many,...
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Cross section of a plane in a cube

A plane is going through a cube so the cross section is a pentagon. Prove that the area of the pentagon is smaller than the product of the pentagon's two largest sides. The first thing I tried to do ...
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Intersection of perpendicular chords: is this true for a sphere?

It is true that in a circle with radius $R$, if the intersection of any two perpendicular chords divides one chord into lengths $a$ and $b$ and divides the other chord into lengths $c$ and $d$, then $$...
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Prove area relations of triangle with giving vector equation

In a triangle $ABC$ with point $I$ inside it such that $\overrightarrow{IA}+2\overrightarrow{IB}+3\overrightarrow{IC}=0$. If $S$ is the area of $\triangle ABC$ and $T$ is the area of $\triangle AIC$ ...
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No right answer for a high school competition-level plane geometry problem?

Let ABCD be a parallelogram with area 15. Points P and Q are the projections of A and C, respectively, on to line BD; points R and S are the projections of B and D, respectively, on to line AC. See ...
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Construction of a rhombus with given conditions

given two parallel lines $P$, $Q$ and two points $X$,$Y$ how to construct a rhombus $ABCD$ passing through $X$,$Y$ and opposite sides lie on $P$ and $Q$. I solved a special case of this problem where ...
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Find orthogonal vector that is in plane

The equation of the plane is $\pi: -x + 2y - z = 2$. I need to find a support vector that is orthogonal to the plane and is also in the plane itself. To me, that sounds contradictory, because if the ...
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Circle Sector Area Calculation - I am getting wrong answer

I am trying to calculate the area of a circle segment using MS-Excel. The radius value is stored in variable "rr", the angle is stored in variable "a". There are 2 equivalent ...
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Definition of a Vertex

While working with a student today, I had the unfortunate realization that I do not actually know a rigourous definition for a vertex (here in the sense of plane geometry). It didn't stop us from ...
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There is a plane given the equation. If a sphere is tangent to this plane, what is the equation for the sphere?(Sphere Center: ${M(4,-2.3)}$)

This is plane equation: ${-2x+y-2z+7=0}$ What is the equation for the sphere tangent to this plane? Center of the sphere = ${M(4,-2,3)}$ Sphere equation form: ${(x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r^{2}}$ ...
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Does $\mathbb{R}^2$ Contain Uncountably Many Disjoint Copies of the Warsaw Circle?

The Warsaw Circle is defined as the closed topologist's sine curve, with an additional arc attached at its free end point and one of the end points of the critical line: Since we don't have an ...
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Find the condition in which $\angle ADB = 3\angle BAC$

See below in acute triangle $ABC$, $D$ is on $AC$ such that $AD=BC$. $CF$ is the angle bisector of $\angle ACB$. $DE \parallel CF$. $E$ is on $AB$. $AE = CD$. Prove that $\angle ABC = 2 \angle BAC$. ...
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Joining faces in planar graph

Let $G=(\{1,\ldots,n\},E)$ be a conncected graph which is planar in the embedding where the vertices $1,\ldots,n$ are placed equidistantly on the circle and all edges are drawn as straight lines. This ...
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vector from a point P to a point Q in a plane in the direction of the normal vector

I'm a little confused with a task in my textbook: Plane $A$ has equation $r*n=k$ (scalar product form) and Point $P$, outside $A$, has position vector $\vec p$. a) Write down a vector equation of the ...
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Construct an affine transformation given the image of 2 points without skewing

I am attempting to create a mobile app in which the user can interact with content on the screen by using two fingers to translate, scale and rotate the content - very much like the interaction with ...
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Why one cannot equate a sphere and a plane but a sphere with a sphere? [closed]

This is a general question about the intersection of a sphere with a plane or sphere which is confusing me. To find the intersection between two spheres K1 and K2, you can equate them, solve the ...
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problem involving centroid and perpendiculars

in triangle $ABC$, $G$ is the centroid and $L$ is an arbitary line through the centroid. $K,I,J$ are the foot of the perpendiculars from $A, B, C$ i need to prove that $KA+JC=BI$ how to use the ...
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If two convex polygons tile the plane, how many sides can one of them have?

The set of convex polygons which tile the plane is, as of $2017$, known: it consists of all triangles, all quadrilaterals, $15$ families of pentagons, and three families of hexagons. Euler's formula ...
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Find the equation of a plane containing line $L$ and equidistant from two points

I want to find the equation for a plane that is equidistant from the points $P_0 = (1, 2, 3)$ and $P_1=(3, 2, 1)$ and contains the line $\ell=(1, 1, 1) + t(0, 1, -1).$ I know that the midpoint of the ...
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Finding the angle of a line in 3D space projected onto a 2D plane

I'm actually a programmer here. I have a math problem, but I don't major in math so I may not explain this clearly. I have a line in a 3D space with points A and B (e.g. (3,4,8) to (6,9,2)). I (or my ...
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A question about the oriented angle of two lines

We know that if $d_1:a_1x+b_1y+c_1=0$ and $d_2:a_2x+b_2y+c_2=0$ then the angle between $d_1$ and $d_2$ is the angle between their normal vectors $\vec{N_1}=(a_1,b_1)$ and $\vec{N_2}=(a_2,b_2)$ and we ...
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What does the vertical plane which passes through the the line of the greatest slope of an inclined plane represent?

I encountered a problem while solving a question about friction force regarding static bodies The Question is as follows, "A body of weight (W) is placed on a rough plane inclined to the ...

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