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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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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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How can I solve $x^2+y^2-z^2=0$? [on hold]

In a Computer Science exercise I am doing, each of $x, y$ and $z$ take integer values between 1 and 50. How can I know all the values each variable take? The result is going to be a list of tuples of ...
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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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1answer
53 views

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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2answers
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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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2answers
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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 ...
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Geometry problem (proving a relation between sides of a triangle) [duplicate]

Let $ABC$ be a triangle with unequal sides. The medians of $ABC$, when extended, intersect its circumcircle in points $L, M, N$. If $L$ lies on the median through $A$ and $LM = LN$, prove that: $2BC^{...
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Light rays in a prism

While studying about path of light rays while passing through a prism, I noticed that: Although the prism is a 3d object, only a cross section of the prism is considered enough to talk about light ...
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Prove that $\frac{BC}{AH}×\frac{CA}{BH}×\frac{AB}{CH}$ = $\frac{BC}{AH}+\frac{CA}{BH}+\frac{AB}{CH}$, where $H$ is the orthocenter of $\triangle ABC$

In $\triangle ABC$, three altitude lines $AE$, $BF$ and $CD$ dropped from three different vertex point intersect one another at point $H$ and $H$ is its orthocenter. Prove that $$\frac{BC}{AH}×\frac{...
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Complex number vs Complex exponential

I know that a complex number is a point in 2D plane. I wonder how to describe what is a complex exponential?
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Find sufficient conditions in which the triangle $ABC$ is located inside the domain traced by the triangle $DEF$.

Let us consider two triangles $ABC$ and $DEF$ in the plane. My question is: Find sufficient conditions in which the triangle $ABC$ is located inside the domain traced by the triangle $DEF$.
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Find the value of angle using elementary geometry rules

In $\triangle ABC$ with base $AC$, $\angle C$ = $46^\circ$ and $AC$ is extended to point $D$. $E$ is a point on $AB$ and $DE$ is joined. Given that $AB=AD=DE$. Find $\angle ABC$.
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Find equation of plane from 2 reference systems

Consider in $R^{3}$ two reference systems $S= (O; \vec{e_{1}},\vec{e_{2}},\vec{e_{3}})$ and $S'= (O'; \vec{e_{1}}',\vec{e_{2}}',\vec{e_{3}}')$, with $\vec{OO'}= (−2,3,9)$ in $S$, $\vec{e_{1}}'=\vec{...
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3answers
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Find the value of $CE$. [closed]

$ABCD$ is a square of side 1, the value of $FE$ is 1 and the points $A$, $C$, $E$ are collinear, as well as $B$, $F$, $E$. The question is to find the value of $CE$. My teacher gave me this challenge ...
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In $\triangle CDF$, $CE$ $FB$ and $DG$ are placed in such a way that they intersects at point $H$. What is the value of $\frac{CH}{HE}$?

SOURCE: Bangladesh Math Olympiad 2014. The inner circle of $\triangle CDF$ touches $CD$, $DF$ and $FC$ at $B$, $E$ and $G$ points respectively. $CE$, $FB$ and $DG$ meets at the point $H$. The side $...
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1answer
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In $\triangle ABC$, $D$ is an exterior point such that $AC = CD$ and $CE$ is parallel to $AF$. Find the area of $ABDF$.

In $\triangle ABC$, $CB$ is extended upto $D$ so that $AC$ = $CD$. An angle $\angle DCE$ is drawn at point $C$ so that is equal to $\angle CAB$ and $AB$ meets $CE$ at $I$.$E$ is such an external point ...
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1answer
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How to define the Cartesian plane in MATLAB

I want to define a small plane in which 'x' changes from (0.5 to 49.5), 'y' changes from (0 to 4.8) and 'z' changes from lets say (2.5 to 4.4). Its like a small piece of a pitched roof placed at an ...
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1answer
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Let $\triangle{ABC}$ be a triangle with area $120$ such that $AB=4AC$. Let $P$ be the intesection between median $CM$ and angle bisector $AD$.

What is the area of $\triangle{PCD}$? I have tried using mass points, and this is where it has gotten me: $x=AC, 4x=AB.$ $\frac{x}{DC}=\frac{4x}{DB} \rightarrow DB(x)=4x(DC)$ Dividing both sides ...
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1answer
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How to calculate s1 and s2 for vector additions

I have attached the question i need help with. I have solved from part a to part c, but dont really understand how to workout part d and part e, can someone please explain what those equations mean ...
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0answers
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Prove without calculus: 2 tangent segments to convex curve longer than curve

Consider a convex curve in the plane. Let B and C be any two points on it, and let A be the intersection of the tangent to the curve at B and C. I would like to show, without calculus, that $AB + AC &...
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Does there exist a partition of an L to create a square?

Background: Consider the following collection of tiles. These can be arranged to form a "difference of two squares" which I call an "L" (shown above), or a "square" (shown below). In this particular ...
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Geometric interpretation of the equation for a plane

So I have to answer the following question: Let n be a unit vector and let Π be the plane with equation r · n = d. Give a geometric interpretation of the number d. [Hint: Consider the formula for the ...
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1answer
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Identifying the square $DEFG$ and than finding the value of its perimeter.

Let $ABC$ be a triangle and $DEFG$ be a square, where $D, E$ points are located on $AB$ and $AC$ or their extension line. $F, G$ points are located on $BC$ or the extension of $BC$. The perpendicular ...
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1answer
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How to rotate a plane around a line that lies on it by alpha degrees?

I am not much of a mathematician and I hope I could get some help with my problem. I have two points in 3D and they form a line. Using 3D programming I am slicing the volume with a plane. All I need ...
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1answer
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Divide a plane with $2n$ points into two equal halves

How can we divide a plane with $2n$ points into two equal halves with $n$ points each using a line?
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2answers
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Proving $|CH| = 2|OM|$ of $\triangle ABC$ of which $O$ is the circumcenter and $H$ is the orthocenter.

Let $ABC$ be a triangle as shown in the figure below, where $O$ is its circumcenter and $H$ is its orthocenter. $AB$ is the opposite side of the climax point $C$, and $OM$ is perpendicular to $AB$. ...
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1answer
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Finding two diametres |$AC$| and |$AD$| where $B$ is the center of the larger one and both the circle touch the point $A$.

Two circle as shown in the figure, A is the tangent point of both the circle. B is the centre of the large circle. The distance of CD = 90 mm(according to estimation) and EF = 50 mm. What is the value ...
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1answer
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What is the value of $ \frac{AH}{AD}+\frac{BH}{BE}+\frac{CH}{CF}$ where H is orthocentre of an acute angled $\triangle ABC$.

SOURCE: SAMPLE QUESTION OF BD MATH OLYMPIAD. In an acute-angled triangle ABC, AD, BE and CF are respectively perpendicular to the opposite side of the three climax point included A, B and C. H is ...
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Need some help with analytic geometry

Check if the following point lies on the plane π : 2x − 3y + 4z − 5 = 0 (point) A = (1, -1, 0) Check if the following vector lies on the plane π : 2x − 3y + 4z − 5 = 0 (vector) -AC = (1, 2, 1) ...
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1answer
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Formal definition of “planar graph”

The wikipedia definition of "planar graph" says: In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges ...
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Functions that map a quadrilateral to the unit square?

Given some quadrilateral $Q \subset \mathbb R^2$ defined by the vertices $P_i = (x_i,y_i), i=1,2,3,4$ (you can assume they are in positive orientation), is there a function $f: \mathbb R^2 \to \mathbb ...
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1answer
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Incircle of a triangle

In the above image, it says $$AE = \frac{bc}{c+a}$$ and $$AF = \frac{bc}{a+b}$$ But $AE$ and $AF$ are tangents from $A$ to the incircle. As tangents on a circle from a given point are equal, $AE=AF$ ...
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a problem about the incircles of two triangles that the orthocenter formed.

See below diagram. $H$ is the orthocenter of an acute triangle $ABC$ where $AB \neq AC$. The circle centered at $I$ and the circle centered at $J$ are the incircles of triangles $ABH$ and $ACH$. $XY$ ...
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1answer
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Chord partition of regular polygon: same fraction of area and perimeter?

This is a variation of a question posed by James Tanton on Twitter. Let $P$ be a regular $n$-gon, $n \ge 3$. A chord $c$ of $P$ is a segment connecting two distinct points of the boundary of $P$, on ...
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2answers
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Do the given perimeter and area corresponds to many shapes? [closed]

I have a perimeter P and area A of a planar shape. How to prove that there are many shapes that corresponds to those perimeter and area values?