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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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rotate plane according to pitch,yaw and roll

Lets say, i have ground-plane equation = $ax + by + cz + d$ . Then, i rotate camera and i know new yaw ( $\theta$ ), pitch ( $\alpha$) and roll( $\gamma$) angle of camera. How can i calculate new ...
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Plane rotated about line of intersection to another plane?

The plane ax+by=0 is rotated about its line of intersection with the plane z=0 through an angle n. What is the equation of the plane in its new position? I saw a question in stack exchange relating to ...
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How to find the plane which contains a point and a line

I know that $\Pi$ contains the point $(2,0,5)$ and the line $\frac{x-10}3 = \frac{y-3}2 = \frac{z-7}2$. How would I find the minimum vector connecting the point and the line so I can then work out the ...
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Given a circle $A$ of area 1 centered at $\{0,0\}$, give conditions that another circle $B$ of known area <1, lies totally within $A$

Given a circle $A$ of area 1 centered at $\{0,0\}$--so, of radius $\frac{1}{\sqrt{\pi}}$--give conditions on the possible location of the center $\{x,y\}$ of another circle $B$ of known area $\pi r^2 &...
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How to intersect arbitrary 3d plane (with form ax+by+cz + d = 0) with the XY plane?

So I know there have been other stack exchange questions around intersecting two arbitrary 3d planes like here (Plane-plane intersection in python) however I have a special case where I want to ...
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Proving plane geometry problem

Given 8 lines on a plane and no two of them are parallel. Prove that, at least two of them form an angle less than 23°. I have checked this out using different angles and the statement seems to be ...
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Why are parallelograms defined as quadrilaterals? What term would encompass polygons with greater than two parallel pairs?

It seems the definition of a parallelogram is locked to quadrilaterals for some reason. Is there a reason for this? Why couldn't a parallelogram (given the way the word seems rather than as a ...
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What is the condition that allows a triangle to be enclosed between two homothetic hexagons?

Let us suppose, as on the figure, that I have a (given) hexagon $DEFIHG$. Then, I build the hexagon $KLMNOP$, which is homothetic of ratio $\alpha$ of the first one ($\alpha$ is a parameter). That is, ...
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How to get the point of reflection given two points and a plane?

I'm working on a 3D simulation, and I need to find two vectors: 1. from the viewer to a point on a water plane, and 2. from the point on the water plane to the sun. What I know is that I can use ...
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Manhattan distance problem with infinite zig zags

If you turn left/right any finite number of times going from point to point, it will be the same as if you traveled $x$ then turned once and traveled $y$ to get there. I hear that even an infinite ...
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Reflecting a plane in another plane

so I'm working on a past paper for my maths exam and there are no provided answers. I am given two planes $2x + 4y -4z = 22$ and $\begin{pmatrix}-1\\3\\2\end{pmatrix} + s\begin{pmatrix}1\\-1\\-1\end{...
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Find a general form of a subplane

I am trying to find a general form of a subplane $ x=2+5t$ , $ y=3$, $ t \in \mathbb{R}$.I know how to find a general form of a line, but not a plane. Could you help me? Thanks!
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A geometry problem : A circle, centred at $I$ has diameter less than the length of the segment. [closed]

Let $AB$ be a line segment with midpoint $I$. A circle, centered at $I$ has diameter less than the length of the segment. A triangle $ABC$ is tangent to the circle on sides $AC$ and $BC$. On $AC$ a ...
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Reconstructing a polygon from the Midpoints of Its Sides

I was reading through Dijkstra's 'A Collection of Beautiful Proofs' and stumbled upon this elegant piece of work: 11. Reconstructing an odd polygon from the midpoints of its sides. We shall ...
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Efficient way to which is the point in between of three points [closed]

I have three points $(x_1, y_1),~ (x_2, y_2),~ (x_3, y_3)$ that are on the same line. How to efficiently find which is the point in between. Example Also, is there any efficient way to check if 3 ...
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Calculating the velocity of a particle on a spinning disc

The particle is moving towards the center as the disc is spinning, the position of the particle is described by the following expression: $r = r(t) cos(ωt)i + r(t) sin(ωt)j$ How do I calculate the ...
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Can we fill the plane with a certain operation?

Background: Paint the origin $(0,0)$ black in $\mathbb{R}^2$. Let $S$ be a set $\{ (x,y) \in \mathbb{R}^2 ~|~ x^2 + y^2 =1 \}$. Paint $S$ black. Paint $(u,v) +S$ black for all $(u,v) \in S$. (...
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stability of $(0,0)$ for $\dot{\theta} = y$ and $\dot{y} = -\sin\theta$

Given the system $\dot{\theta} = y$ and $\dot{y} = -\sin\theta$. For the fixed point $(0,0)$ I can see through linearisation that the flow corresponds to a centre which moves anticlockwise Why is ...
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which statement is true and why? [closed]

Let A and B be two distinct points in the plane, d their distance apart, and r a given positive integer. Then (A) there always exists a circle of radius r passing through A and B (B) if d ≤ 2r then ...
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How to find the value of the projection of an angle onto a plane?

Consider a triangle (T) in 3D space of given vertices A, B, and C. A given ray (R) (assumed to be in the direction of the x-axis) hits the triangle at one of its vertices - say A. Let $\theta$ be ...
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Distance between a point and a line in space with unknown line equation

We have $A(-2,3,1)$ and we have to find the distance from $A$ to line which contains point $P(-3,5,2)$ and this line makes equal points with coordinate axis. I know how to solve this, I need the ...
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What's the projection of the $x + 5y + z = 4$ plane onto the $xy$ plane

The vector field: $$F = 10xy^2 \hat{i} + (4z-2xy^2) \hat{j} + (5y -2x^2y) \hat{k}$$ The line integral yields $\int_{c} F \cdot dr = \frac{230}{3}$, where $c$ is the curve it goes along. It's just ...
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Direction Ratio of line when one of contributing plane is constant

A line is given by equation $$x-5=5-y, z=5$$ Is the direction ratio of the line 1,1,5? As per my reasoning all points on the line are its direction ratio, so a,b,5 will the direction ratio provided ...
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Equation of plane containing a point and a line

Find the equation of the plane containing the point $A(0,1,-1)$ and the line $(d) : \begin{cases} 2x - y + z + 1 = 0 \\ x + y + 1 = 0 \end{cases}$ Where should I start? I was thinking about writing ...
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Barycentric and projective coordinates

I have a question: what is the relation between the barycentric and the projective coordinates? Are the first one a particular case of the second? Thank! Edit: the setting is the plane, and in ...
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Circular inversion notes or books

I am preparing a small lesson that I have to do about geometry. The topic of the lesson is circular inversion. Can you suggest me some sources where I can find some nice material about it? Thanks!!
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Triangle inscribed in a square

Consider a square $ABCD$ of side length $1$. Have the intersection of the diagonals $AC, BD$ be $G$. Determine the value of largest (in terms of area) inscribed triangle, such that $G$ is the ...
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Derive distance formula between two parallel planes [duplicate]

Given two parallel planes: $$ ax + by + cz = d \\ ax + by + cz = e $$ How would I derive a formula for the distance between them?
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Exactly one of the three points $L,M,N$ lies on the triangle $ABC$, where $AL, BM, CN$ are proper Cevian lines

$AL, BM, CN$ are proper Cevian lines and are concurrent at an ideal point. To prove - Exactly one of the three points $L,M,N$ lies on the triangle $ABC$. I was thinking that from Ceva's theorem we ...
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How can $2x -5y +z = 3$ be equation of a line?

The equation of the plane containing the lines $2x - 5y + z = 3$, $x + y + 4z = 5$ and parallel to the plane $x +3y + 6z = 7$? I actually have the solution to this question, but I don't understand it. ...
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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 ...
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Optimal covering with $n$ non-necessarily equal discs

What kind of algorithm can I use to search for an optimal (minimum area) covering of a limited region of the 2d plane with $n$ discs $(x_i, y_i, r_i)$? I've found many investigations on fixed radius ...
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Rational distance problem

My question is related to kind of problems, called "rational distances problem"(at least by wolfram mathworld). I couldn't find a specific solution, so it would be a real help if you have an idea or ...
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locus of foot of perpendicular in 3 d Geometry

A variable plane cut the coordinate axis at $x,y,z$ axis at point $A,B$ and $C$ respectively such that the volume of Tetrahedron $OABC$ is remain constant and equals $32$ cubic units and $O$ represent ...
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Using Ptolemy's Theorem to find length ratio

In this figure, $X, Y$ are tangent points and $\frac{DX}{EX} = \frac{8}{3} , \frac{EY}{DY} = 4 , \frac{AC}{AB} = \frac{5 }{4} . $ Then, what is $ \frac{BC}{AX}$ ? System of equations from the ...
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Can 3D co-ordinates be transferred into 2D co-ordinates?

Is it possible to transform co-ordinates $(a,b,c)$ into $(x,y) $ such that $(x,y)$ is unique for each $(a,b,c)$ ? $a, b, c, x, y$ are in $\Bbb{R}$ .
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Given $n$ circles of radii $r_1,r_2,…,r_n$ inseparable by straight lines, prove that they can be covered by a circle of radius $r_1+r_2+…+r_n$

Definition: A subset $A\subset\mathbb R^2$ is inseparable by straight lines if there doesn't exist a straight line $L$ such that $L \cap A=\emptyset$ and $L$ divides $A$ into $2$ nonempty parts, ...
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What is the name of a plane with boundary?

A plane by definition does not have boundary. When we are taking a connected subset of a plane, it is planar everywhere. Can we call it a "plane with boundary"? What if it is an open set? Are ...
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How many straight lines and circles can be drawn in the plane so that they are equidistant from all four points?

Four points on a plane are given which are not collinear or all on one circle. How many straight lines and circles can be drawn in the plane so that they are equidistant from all four points? If not ...
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Finding the area of inner triangle constructed by three cevian lines of a large triangle

QUESTION: In a triangle $ABC$, $AD, BE$ and $CF$ are three cevian lines such that $BD:DC = CE:AC = AF:FB = 3:1$. The area of $\triangle ABC$ is $100$ unit$^2$. Find the area of $\triangle HIG$ ...
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How many lines are represented when only two direction cosines are given

If the cosine angle is given for only X and Y axis, but missing (not mentioned) for the Z axis. How many lines can be represented by the two given direction cosines. The text book say two, one forming ...
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Given three non-overlapping circles, find the triangle of minimum perimeter with one vertex on each circle

G. Polya "Mathematics and plausible reasoning" Chapter 9, problem 2: Three circles in a plane, exterior to each other, are given in position. Find the triangle with minimum perimeter that has one ...
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1.a) A plane $S_{1}$ contains the three points $(1,0,0),(0,1,0)$ and $(0,0,1) .$ Find an equation for $S_{1}$

1.a) A plane $S_{1}$ contains the three points $(1,0,0),(0,1,0)$ and $(0,0,1) .$ Find an equation for $S_{1}$ 1b) The perpendicular from the origin $\mathrm{O}$ to another plane, $\mathrm{S}_{2}$ ...
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Condition for three mutually perpendicular planes in 3D Geometry.

Let us consider three mutually perpendicular planes $l_ix+m_iy+n_iz=p_i$ for $i=1,2,3$, where $l_i,m_i,n_i$ are direction cosines of the normals to the planes. Since these three planes are mutually ...
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Find a plane that goes through three given points

The Question Data is collected on a person’s income (thousands of dollars), their age, and the value of their home (thousands of dollars). We would like to predict home value, H, as a function of ...
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Finding $\frac{PQ}{QR}$ in a right angled $\triangle ABC$, where $AD$ is the median line dropped from the opposite vertex of the hypotenuse

Let $\triangle ABC$ be a right angled triangle where $\angle A = 90^\circ$. $D, F, E $ and $G$ are the midpoints of $BC, AB, AF$ and $FB$ respectively. $AD$ interesect the lines $CE, CF$ and $CG$ at ...
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Finding the ratio of a side of $\triangle ABC$ and its segment where one cevian line from the opposite vertex intersect the side in any point

In $\triangle ABC$, $L$ and $M$ are two points on $AB$ and $AC$ such that $AL = \frac{2AB}{5}$ and $AM = \frac{3AC}{4}$. $BM$ and $CL$ intersect at the point $P$ and the extension line of $AP$ and the ...
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Medians of ABC, when extended, intersect its circumcircle in points L,M,N. Prove that if $LM=LN$ then $LM=BC$

Here is the original problem that I was able to solve here: The medians of $ABC$, when extended, intersect its circumcircle in points $L, M, N$. If $L$ lies on the median through $A$ and $LM = LN$, ...
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R²/Plane Subset Equation With Plane Homothetic Transformation

Let's consider $H_k∶\ \left\{\begin{matrix}\mathbb{R}^2\rightarrow\mathbb{R}^2\\(x,y)\longmapsto(kx,ky)\\\end{matrix}\right.\ $. It is an homothetic transformation of $\mathbb{R}^2$ of center $(0,0)$...
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Book on bogus proofs or counterexamples in plane geometry

There are books about counterexamples in analysis, topology or probability. Is there any book that focuses on counterexamples in plane geometry or loopholes in geometrical proofs? I am particularly ...