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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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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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1answer
50 views

A small problem about the circumcentre and orthocentre of a triangle.

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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Determining the value of AH\AD+BH\BE+CH\CF where H is orthocentre of the three diagonals AD, BE and CF in an acute-angled triangle ABC.

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 the orthocentre of the orthogonals. What is the ...
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2answers
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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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538 views

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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2answers
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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
39 views

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
38 views

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
75 views

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?
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4answers
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Tangent plane to $x^2+y^2+z^2=50$ at $(3,4,5)$

Prove that the tangent plane to $x^2+y^2+z^2=50$ at $(3,4,5)$ is $3x+4y+5z=50$ My workings are shown below but get the answer wrong completly, have I made a simple mistake is my method flawed. So if ...
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4answers
41 views

Find the equation of the tangent plane to $xy+yz+zx=11$ when $x=1$ and $y=2$

Find the equation of the tangent plane to $xy+yz+zx=11$ when $x=1$ and $y=2$ giving the answer in the form $f(x,y,z)=k$, where $k$ is a constant and $k\in \Bbb{Z}$. So I know that the tangent plane ...
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1answer
182 views

Problem of three circles

This geometrical problem was proposed in a Mathematics Contest for high school students of my country. It is truly hard to find its solution. Let $ABC$ be an acute triangle inscribed in the circle ...
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4answers
187 views

Sum of squared inner product of vector with spokes around unit circle is constant

Let $v$ be any vector in the plane, and $\{w_i\}$ be $n>2$ vectors evenly spaced around the unit circle. Then it seems true that $$\sum_{i=1}^n (v\cdot w_i)^2 = k \|v\|^2$$ where $k$ is a constant ...
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2answers
54 views

How to draw this function?

I have a problem where I am given this function: $z(x, y) = \cos(x)\cos(y)^{T}$ and now I am supposed to determine if a point is above or below this plane. But I am having trouble understanding ...
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0answers
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A Problem With Coordinate Systems

Consider a coordinate system $\cal{C}$ such that the concentric half circles around two fixed points $P_1,P_2$ in the plane above line $P_1P_2$ create the grid. So any point in the upper half plane in ...
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2answers
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Find the equation of the plane containing these lines

So I was given these lines: $$\frac{x-1}{2} = \frac{y-2}{-2} = \frac{z}{-1}$$ $$ \frac{x}{-2} = y+\frac{5}{3} = z-\frac{4}{0} $$ And I was asked to find out if they are parallel or perpendicular. Now ...
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1answer
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Given a point $x$ inside the triangle $ΔABC$, prove that $XA+XB < CA + CB$

Given a point $x$ inside the triangle $ΔABC$, prove that $XA+XB < CA + CB$ I have used triangle inequality in every way possible, nonetheless, I haven't come up with a proof, since I always get ...
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1answer
13 views

Detect Crossed Paths on a Plane Given Coordinates.

If you have the $X$ and $Y$ coordinates for $2$ lines on a $2D$ plane/graph is it possible to detect via true/false if they cross paths? I'm just trying to detect if lines are not parrallell. If $(X,...
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1answer
21 views

Finding planes intersecting along a specific line

I have a plane $V$ (it's basis) in $\mathbb{R}^3$ and a vector $a$ that belongs to that plane. What should I do when I want to find all planes that intersect $V$ along the line created by $a$? How ...
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2answers
27 views

Proof regarding altitudes of a triangle and a midpoint of one of its sides

Let $\triangle ABC$ be a triangle with altitudes $\overline{AH}$ and $\overline{BK}$. Consider the axis of the segment $ \overline{HK}$. Let $M$ be the point of intersection between the axis and the ...
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0answers
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Is there proof of equation $P = L - I + 1$ in the $2D$ space from the computational geometry?

QUESTION: Does exist a proof of statement $ P = L - I + 1$, where $P$ stands for number of polygons, $L$ for number of line segments, $I$ for number of intersection points, in any arrangement made ...
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Angle between two intersecting lines that appear on a cut face

I am given the equations of 2 thin planes in a rock. Plane 1: $\mathbf r\cdot(0,3,-1) = -1$ Plane 2: $\mathbf r\cdot(1,0,1) = 1$ The entire set (the rock) is cut by the plane: $\mathbf r\cdot(-1,3,...
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3answers
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How to prove the triangle is isosceles and determine its area

Let $P(a,b)$ be a point on the curve $y = \frac{1}{x}$ in first quadrant and let the tangent line at $P$ intersect the $x$-axis at a point $A$. Show that triangle $AOP$ is isosceles and determine its ...
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1answer
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Describe the shape of projection of vertices (vector positions of a cube) onto a 2D plane from a source (position vector)?

I am having trouble with this. I can manually calculate every single projection point onto the z=0 plane from deriving vector equations to get to the z plane for each vertices. From this I can then ...
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0answers
19 views

How can prove that three circles coaxial?

Let $(O_a)$ meets $(O'_a)$ at $A_1$, $A_2$; $(O_b)$ meets $(O'_b)$ at $B_1$, $B_2$; $(O_c)$ meets $(O'_c)$ at $C_1$, $C_2$ such that $A_1$, $A_2$, $B_1$, $B_2$, $C_1$, $C_2$ lie on a circle. Let $(O_a)...
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1answer
15 views

Finding parameters for which the line lies in the plane

I tried to solve the following task: A line L has equation: $\frac{x-2}{p} = \frac{y-q}{2}= z-1$, where $p,q \in \mathbb{R} $. A plane P has equation: $ x +y +3z = 9$. Given that line L lies in the ...
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finging the formula for the radius of the cross section relation/ratio to x-cordinate dependent on the cuting plane

"banana shaped" body is located between two planes that cross with x-axis $ x=7 $ and $ x=-7 $. Cutting the body with the planes that cross with the form circles, that diameter that endpoints are ...
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2answers
49 views

Determining whether a large series of 3D points all line on a plane

TL;DR: For a large series of precise 3D coordinates that describe a real-world orbit, how can we determine if they all lie exactly on a plane? The Problem I've used NASA's highly precise SPICE ...
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1answer
64 views

Internal angles in regular 18-gon

This (seemingly simple) problem is driving me nuts. Find angle $\alpha$ shown in the following regular 18-gon. It was easy to find the angle between pink diagonals ($60^\circ$). And I was able to ...
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1answer
32 views

Finding the angle between a line and plane exercise

The aim is to find the angle between a line and plane and if they intersect then the point where they do that. given: $$ \frac{x+1}{2}=\frac{y-3}{4}=\frac{z}{3}, 3x-3y+2z-5=0 $$ What I have found ...
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0answers
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Plane Geometry problem: $a(PA)^2\ +\ b(PB)^2\ +\ c(PC)^2$ is minimum

Problem: Let $ABC$ be a fixed triangle and $P$ be a variable point in the plane of $\Delta ABC$. If $a(PA)^2\ +\ b(PB)^2\ +\ c(PC)^2$ is minimum, then the point P with respect to $\Delta ABC$ is (A) ...
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1answer
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Linear algebra: Analytic geometry problem.

Let $p$ be a line given by the equation $x-1=\frac{y}{2}=\frac{z+3}{2}$, and $q$ a line with the equation $\frac{x}{2}-1=y-2=\frac{z+1}{2}$. If we reflect the $p$ over the plane $\Pi$ we get the line $...
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1answer
35 views

How many points are there in the following set? [closed]

Let us consider the following set: $A=\{(x, y, z) \in \Bbb{R}\times\Bbb{R}\times\Bbb{R} : ax+by+c=0,z=0 \},c\neq 0$ and $B=\{(x, y, z) \in \Bbb{R}\times\Bbb{R} \times\Bbb{R} : ax+by=0,z=0\}$. Then ...
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1answer
64 views

$n$ lines in a plane, proper coloring of intersection points with just 3 colors

Draw $n$ lines in a plane so that there are no parallel lines and there are no three lines passing through the same point. Each intersection point is colored red, green or blue. Prove that it is ...
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Vector addition exercise: Plane and Wind

I sit since 2 days on it and can't solce it: A plane flies with the speed of vF=240 km/h in direction of north. It flies into a storm from north east with the wind speed of v=90 km/h. What's the ...
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5answers
31 views

Intersection of a cylinder and a plane

Given a plane: $$x - y + z = 0$$ And a cylinder $$x^2 + y^2 = 2$$ Why can't I get the intersection of the two by equaling both equations? i.e. $$x^2 + y^2 - 2 = x - y + z \implies x^2+y^2 -x + y - z ...
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0answers
28 views

Calculate X, Y, Z of the 4 points in the 3D space.

I want to find the $x, y, z$ coordinates for $4$ points in a 3D space. Point $A$ is my origin $(X, Y, Z = 0,0,0)$ and other points $B, C, D$ are with reference to point A. I know all six distances ...
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2answers
40 views

The point of intersection of the graph of a quadratic function and a circle

Here is a question I found on the website of International Kangaroo Maths Contest. The question goes like this: A quadratic function $f(x)=x^2+px+q$ is such that its graph intersects the x and y ...
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1answer
19 views

Length and perimeter

Hello guys? I have been having constant disagreements with my fellow professors on this question. A field was to be fenced using 816 posts placed 4 meters apart,leaving a 4 meter space for the gate....
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0answers
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Does the normal of any plane pass through every single point in space? If so, why?

The normal is a direction vector that is perpendicular to (edited) the plane at all points. Is the normal then able to pass through every single point in space? Edit: Suppose I can shift the normal ...
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1answer
55 views

Find angle $ \angle AED $ in the following triangle. [duplicate]

Find angle $ \angle AED $ in the following triangle. In the above triangle we have : $CA=CB ,CE=DB=BA ,\angle ACB =20^° , \angle CAB=\angle CBA=80^°$ now find $ \angle AED $. I think if we draw ...
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1answer
15 views

Determine the slope of a plane whose rise in z is zero, but whose change in x and y are not.

Determine the slope of a plane whose rise in z is zero, but whose change in x and y are not. Explain what this plane looks like. Would it be a plane in the xy plane? Not sure how to start this ...
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1answer
22 views

Self-Similar Polygon Tessellations

It is well-known that the only regular polygons which tessellate the plane (using only one shape) are the triangle, square, and hexagon. However, there are many more tessellations of the plane by ...
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1answer
52 views

How to study Euclidean geometry from axioms?

I want to know if there's a good book or any other type of guide to study Euclidean geometry by only the 5 axioms in plane geometry and prove every other theorems from them?
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2answers
100 views

Number of Intersection points between lower half of an ellipse and a circle

An ellipse has its axes parallel to the coordinate system axes and its major axes is parallel to X-axis. Meanwhile, there is a circle located at the coordinate system origin, whose radius is smaller ...
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1answer
45 views

Solving the equations in three variables

When I am dealing with some geometry problem in barycentric system I come across with the following equations $\frac{x^2}{a}+\frac{y^2}{b}+\frac{z^2}{c}-xy\left(\frac{1}{a}+\frac{1}{b}\right)-yz\...
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2answers
53 views

Geometric proof of equivalence between two constructs of ellipse

Pretending that we don't know any analytic geometry and trigonometry. Consider the following two constructs of an ellipse, where admittedly the second one is an ad-hoc construct for the ellipse ...
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1answer
14 views

Intersection of the two planes

I need help for my vector's assignment!!! Let L be the line of intersection of the two planes x+y+z-1=0 and 2x+3y-z+2=0. Find the scalar equation of the plane that contains the line L and passes ...
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2answers
26 views

Intersection of a plane

I need help for my grade 12 Vector's homework. Can a plane be perpendicular to the x-axis and contain the line x=z, y=0? Explain. I really hope someone can answer this question