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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What is the subspace with scalar equation $-2x_1+3x_2+x_3−x_4=0?$ A plane in $R_4$ or a hyperplane in $R_5?$ [closed]

Problem : The subspace with scalar equation $-2x_1+3x_2+x_3−x_4=0$ is a. a plane in $R_4$. b. a hyperplane in $R_5$. My Thought Process : I think since there are $4$ variables in scalar equation, ...
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Find the area of the triangle $AEZ$ in the figure

For reference: The angle measure $\angle ERZ=75^o$ and $EH=6$. Calculate the area of ​​the triangular region $ZEA$.(Answer:S=4) My progress: $OA =R\\ \triangle OAZ(equilateral)\implies S\triangle OAZ ...
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Find general and parametric equations of the plane containing the points $A(3, 0, 0), B(0, 1, 0)$ 'perpendicular' to the $XY-$plane.

Question : Find general and parametric equations of the plane containing the points $A(3, 0, 0), B(0, 1, 0)$ 'perpendicular' to the $XY-$plane. My Try : Seeing that the plane is perpendicular to the $...
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Ratio of products of line segments

The points $A,B,C,D$ are collinear. The point $P$ sits off the line, and $\angle{APB}=\angle{CPD}=\theta.$ I'd like to show that if the points $P,A,D$ are fixed, the ratio $\dfrac{AB\cdot AC}{DC\...
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We have two convex sets $S,T$ such that $S\subseteq T$. Prove or disprove that the circumference of $S$ must be smaller than that of $T$. [duplicate]

This question originated from an exercise asking to compare the arclength of $y=x^2$ between $(0,0)$ and $(1,1)$ and $\frac{\pi}{2}$. The solution starts by constructing the circle with center $(0,1)$ ...
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Find the radius $r$ of the figure below

Foe reference: Find the radius $r$ of the figure below where $D, F$ and $G$ are points of tangency and $AC = 5$ and $AB = 12$.(S: $r=4$) My progress: $\triangle ABC: BC^2 = \sqrt{5^2+12^2} \therefore ...
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what makes some line to be a tangent line on to other Graph.

I learned tangent line as a concept associated with a circle. I thought to make a tangent line, it might have some specific point which is perpendicular to. as always did with circle did. But in the ...
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Determining A Vector Through the Center of Multiple Points on a Sphere

I am working on a machine vision task that requires me to determine spin rate and spin axis of a moving ball. I have had some luck, and actually do have a solution but am looking for a more efficient ...
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Upper bound for Pach's Selection Theorem for plane

Pach's Selection Theorem for plane says that: Assume $X \subset \mathbb{R}^{2}$ is a finite set of points in general position, partitioned into three colour classes $C_{1}, C_2, C_3$ with $\left|C_{i}\...
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Calculate normal vectors for each element of a grid in Python

How can I quickly calculate the normal vectors of each mesh of my grid? (grid is defined by the three matrices Mx, My & <...
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How to compute the coefficients of the general form of a line $ax + by + d = 0$ in a way that helps to find the plane equation $ax+by+cz+d=0$

I have a bunch of points $(x,y,z)$. I want to compute the best fit line equation going through those points in 2D (I want the line in 2D so I just ignored the z axis for the line). I have done that ...
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Is this question solvable using the law of sines and cosines?

Is it possible to (analytically) calculate the area of the following triangle using the rules of sines and cosines? Is it possible to calculate it using only the rule of sines? The data given is: $BD$...
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As a function of $a$, how many points are there in hyperboloid $x^2 − y^2 − z^2 = 2$ where the tangent plane is parallel to plane $z-ax=3$?

PROBLEM As a function of $a$, how many points are there in hyperboloid $x^2 − y^2 − z^2 = 2$ where the tangent plane is parallel to plane $z-ax=3$ ? MY APPROACH I started by finding the normal vector ...
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Given the period lengths for the orbits of $n$ different planets around the sun, how long until they all align?

Say you have $n$ planets orbiting around the sun, where the $i$th planet takes $t_i\in\mathbb{R}_{>0}$ days to complete one full cycle. Assume at $t=0$, all the planets are aligned with the sun. ...
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Coordinate geometry dividing the plane by pairwise straight line

Three pairwise straight line intersecting (but not at the same point) by how many parts these lines divides the plane?
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Can a circle be drawn to pass through a point and contain a rectangle?

Let $a, b, c, d$ be points in the Euclidean plane. Suppose that $abcd$ is a non-degenerate rectangle, and that the length of the line segment $ab$ is at least as big as the length of the line segment $...
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Can a circle encompassing a continuous, closed plane curve be made to pass through an arbitrary point on the exterior of the curve?

Let $J \subseteq \mathbb{R}^2$ be a continuous, closed curve (closed in the sense that $j(a) = j(b)$ for some continuous parameterization $j:[a,b]\rightarrow\mathbb{R}^2$ of $J$ with $a,b \in \mathbb{...
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Identifying an equilateral triangle in a hexagon. [duplicate]

What is the best way to prove that the midpoints of BC,DE,FA are the vertices of an equilateral triangle given that ABCDEF is a hexagon inscribed in a circle with centre at O if angle AOB = angle COD =...
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Intersection between Segment and Plane

I have to calculate the point of intersection between a segment and a plane. The segment is defined by the two points $S_1=(x_1, y_1, z_1)$ and $S_2=(x_2, y_2, z_2)$. The plane is defined by a point $...
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Intersection between a Plane and a Line (2 points)

I have to calculate the intersection between a line and a plane. Of the line I know two points $P_1=(x_1, y_1, z_1)$ and $P_2=(x_2, y_2, z_2)$ while of the plane I know the equation $Ax + By + Cz + D =...
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Divide triangle into two subtriangles of different area

Let $ABC$ be a triangle and denote its area by $k = \mathrm{area}(ABC)$. I want to divide $ABC$ into two sub-triangles $ABE$ and $AEC$ such that $\mathrm{area}(ABE) = t$ and $\mathrm{area}(AEC) = k-t$...
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Construct a map from plane coordinates to 3D coordinates

Let $u = (u_x,u_y,u_z)$ be a point in $\mathbb{R}^3$. The equation of the plane $P$ passing through $u$ and perpendicular to a unit vector in the direction of $u$ is $$ u_x(x - u_x) + u_y(y - u_y) + ...
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Extending the diagonal of square formula into the Pythagorean theorem?

It came to my mind if we could produce a yet another proof for the Pythagorean theorem by begin with the speacial case of a square. That is, we know that the diagonal of a square is $d=\sqrt{2}a$, ...
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Plane Geometry about ratio of areas

Point $F$ lies on the straight line $AC$, with which a circle centered by $F$ can be made through both $B$ and $C$. $M$ is the midpoint of the arc $BC$. The straight line $CM$ intersects the straight ...
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3 answers
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Find $k\in\mathbb{R}$ given $w = k+i$ and $z=-4+5ki$ and $\arg(w+z)$

I am working on problem 2B-11 from the book Core Pure Mathematics, Edexcel 1/AS. The question is: The complex numbers $w$ and $z$ are given by $w = k + i$ and $z = -4 + 5ki$ where $k$ is a real ...
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How to Find a Point when Given the Equation of the Line it's on, Another Point on the Line and the distance between the Two Points?

(12th Grade Calculus Level) Let's say I'm given point A(1,2,4) and a line [x,y,z] = [4,3,9] + t[3,1,5]. I have to find point B which is on the line and is a distance of 5 units away from Point A. What ...
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2 votes
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Square and equilateral triangles

CDEF is a square, CBF and BEA are equilateral triangles. Find angle $x$. What I have found at the moment: $\angle BEF =15^{\circ}, \angle AED =135^{\circ} $ I managed to find the length of $BE$ and ...
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How to construct the centroid of a quadrilateral

I know how to construct the centroid of a quadrilateral as mentioned here. But my question is different from that. We know that if points B,C,G are given in geometry plane and for locating point A ...
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4 answers
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Derive complex number form its position in the plane

Suppose I have a circle of center $s$ and radius $r$ and four points $p1,p2,p3,p4$ (in order) on that circle. What is the formula for the middle point $m$ between $p1$ and $p2$ on the circle? In the ...
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3 answers
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Show that the area of ​triangle $S_{ABC} = R\times MN$

For reference: Show that the area of ​​triangle $ABC = R\times MN(R=BO)$ I can't demonstrate this relationship My progress: $$S_{\triangle ABC} = \frac{abc}{4R}$$ $$S_{\triangle ABC}=\frac{AC\times ...
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What's the length of the segment $HN$ in the figure below?

For reference: In the drawing, $T$ is the point of tangency, $LN || AT$, $OH = 4$ and $LN^2+AM^2=164$. Calculate $HN$. (Answer: $8$) *Both circles have the same radius. Progress: By Stewart's theorem,...
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Is there any kind of algebraic structure on a line in the hyperbolic plane?

There is a very classical correspondence between projective planes and division algebras: given a plane, each choice of three distinct points (zero, one and infinity) on each line determines addition, ...
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Find the segment $BQ$ in the figure below

For reference: Let a circle $\omega$ (not labelled in the graph) centered at $P$ tangent to $AB$, and $T$ is point of tangency. $\angle APB=90^\circ$. Let $K$ (not labelled in the graph) be some point ...
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Signed Distance of a Point to a Hyperplane

Let's define a affine function $f(x) = \alpha^Tx+\alpha_0$ where $\alpha$ is a vector (weights) and $\alpha_0$ is a constant (bias). Also, define a normal vector of the hyperplane as $n=\frac{\alpha}{...
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Find location and normal of a plane based on the relative intersection points from 3 lines

Here is an illustration of the problem I have 3 points in a 3d space, in my example the points are: (0, 0, 0), (26, 0, 0) and <...
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What´s the length of the segment AC in the triangle below?

For reference: In the triangle $\angle A$ is right and $D$ is a point on the side $AC$ such that the segments $BD$ and $DC$ have length equal to $1 m$. Let $F$ be the point on the side $BC$ so that $...
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In a bichromatically colored plane, is it always possible to construct any regular polygon such that all vertices are the same color?

Let there be a plane $P$ where all points $(x,y)$ are colored either $red$ or $blue$. Is it always possible to construct every regular $n$-gon where all the vertices are the same color? Some ...
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Position of a plane that intersects point

I'm trying to find the position of a plane such that it intersects an arbitrary point. A point P exists along a vector M starting at O, a plane intersects P with a normal vector of N. What is the ...
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Are there closed and simple plane curves (Jordan curves) of finite length that are not piecewise $C^1$ curves?

Are there closed and simple plane curves (Jordan curves) of finite length that are not piecewise $C^1$ curves (or $C^1$ curves for parts, this is, continuous curves that are made up by a finite number ...
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What's the ratio between the segments $\frac{AF.BG}{FG}$ in the figure below?

For reference: In the figure below the trapezoid has height $13$ and is inscribed in a circle of radius $15$. Point $E$ is on the minor arc determined by $A$ and $B$, the points $F$ and $G$ intersect ...
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What´s the value of the ratio $x$ in terms of the ratio $R$ in the figure below?

For refrence: If $P$, $Q$ and $T$ are points of tangency, then $x$ in terms of $R$ is My progress: $\triangle JOT: (R+x)^2=x^2+(R+LT)^2\implies\\ R^2+x^2+2Rx = x^2+R^2+LT^2+2RLT\\ x^2+2Rx = R^2+2RLT\\...
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Is this converse results of Varignon's theorem known?

The Varignon's Theorem on Quadrilateral is very well known results of Plane Geometry and we have find the Converse of this theorem on Quadrilateral and generalise this for 2n-sided convex irregular ...
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A 2-dimensional bounded domain's boundary is connected iff the domain contains the interior of every enclosed closed Jordan curve

I'm looking for a detailed proof, or a reference to a detailed proof, of the following theorem. Let $G$ be a non-empty, bounded domain in the Euclidean plane $\mathbb{R}^2$. Then $G$'s boundary is ...
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What's the measure of angle $PCB$ in the figure below?

For reference: In the interior of a triangle ABC, a point $P$ is marked in such a way that: $PC=BC$ and the measure of the angle $PAB$ is equal to the measure of the angle $PAC$ which is $17°$. ...
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What's the measure of the angle $ACD$ im trapezoid below?

For reference: On a rectangular trapezoid $ABCD$ right at A and B. Mark a point $L$ on $AB$ such that: $CL=AL=AD$ and $AC=CD$, calculate the measure of angle $ACD$.(Answer:$30^o$) My progresss: Follow ...
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2 votes
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Wthat's the measure of the $\angle MCB$

For reference: In a triangle $ABC$, the interior cevian $CM$ is drawn, so that $CM=AB$; Knowing that the measure of $\angle A= 30°$ and $\angle B =100°$. Calculate the measure of $\angle MCB$.(Answer:$...
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The boundary of a chain of plane squares

Denote by $\mathbb{Z}$ the set of whole numbers, by $\mathbb{R}$ the set of real numbers, and by $\overline{\mathbb{R}}$ the set extended real numbers $\mathbb{R}\cup\{\pm\infty\}$. We denote ...
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What am I doing wrong to finding the perpendicular distance between a point and a plane?

From this post How to prove that a plane is a tangent plane to a sphere?, I went and attempted to show that $4x+3z+29=0$ is tangent to the sphere $(x+1)^2+(y-3)^2+z^2=25$. However I keep getting a ...
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Show that $x = \sqrt{ab}$ in the figure below

For reference: If $ABCD$ is a square and $BC$ is a diameter. Show that $x = \sqrt{ab}$. Progress: Let $h$ = height $\triangle TPQ$ $$\triangle TPQ \sim \triangle TAD$$$$ \frac{h+a+b+x}{h} = \frac {...
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Find the 3 radius in the picture below

If ABCD is a square show that: $r1=\frac{l}{16}$, $r2 = \frac{l}{6}$ and $r3 = \frac{3l}{8}$ My progress: Euclid's Theorem: $\triangle AO2D\\ AO_2^2=DO_2^2+l^2-2r_2l\implies(l+r_2)^2 = (l-r_2)^2+l^2-...
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