# 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.

794 questions
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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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### 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 ...