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 in a 2D plane.
1,889
questions
0
votes
2
answers
90
views
Intersection of two planes is a straight line [closed]
Let $a_1x+b_1y+c_1z+d_1=0$ and $a_2x+b_2y+c_2z+d_2=0$ be the equations that describe two planes. In my lecture notes, it's written that $a_1x+b_1y+c_1z+d_1=0$ and $a_2x+b_2y+c_2z+d_2=0$ have a ...
- 149
3
votes
0
answers
29
views
Slope of a hyperplane
I was reading a book (Nonlinear Elliptic Equations of the Second Order by Qin Han), and there was something which I didn't really understand. I will try to simply the setting for my question.
Let $a\...
- 2,237
0
votes
0
answers
43
views
Calculate the measure of the arc A in the circle
Twice the measure of the arc $\overline{AB}$ is equal to the measure of the arc $ASB$ and $\overline{KL}=1$. If $\overline{KB}$ is an integer value, calculate the measure of the arc $AB$. (Answer: $...
- 5,971
-1
votes
1
answer
46
views
Find the angle $x$ in the quadrilateral $ABCD$ below
Calculate the angle x in the figure below (Answer:$80^o$)
Is it possible to resolve it just with the data provided? I think I would need auxiliary lines but I couldn't identify it
$\angle A = 180-80 -...
- 5,971
0
votes
1
answer
38
views
three lines exist in $L1, L2, L3$ $L1$ : intersecting the z - axis at $P(0,0,2)$ and does not meet the x-y plane. find area of triangle formed
three lines exist in $L1, L2, L3$
$L1$ : intersecting the z - axis at $P(0,0,2)$ and does not meet the x-y plane
$L2$ : passing through origin and through point P
$L3$: passing through origin and ...
- 431
-1
votes
0
answers
50
views
Find the smallest integer value of the perimeter of the quadrilateral region "ABCD"
Given the squares "ABCD" and "DMNL", such that "DK=3(MK)" and "MN=5". Calculate the smallest integer value of the perimeter of the quadrilateral region "...
- 5,971
1
vote
1
answer
115
views
Calculate the distance from O to AC in the figure below.
If $T$ is a point of tangency, $(OB)(OC)=24$, $TM=MC$ and $AC=8$. Calculate the distance from O to AC.(Answer:$3$)
$\triangle OFT\sim \triangle CFO \implies \frac{CO}{r}=\frac{FO}{FT}=\frac{CF}{FO}\\
...
- 5,971
0
votes
0
answers
21
views
What is the resulting plane of $plane_1 = plane_2$ intuitively
I have been trying to find the equation of the intersection line between two planes, and was trying some things. The first thing I tried was to just put an equal sign between the two plane equation. I ...
1
vote
0
answers
21
views
Calculate segment of 3D line within distance from 3D triangle
Given an arbitrary 3D line (infinite length) and an arbitrary 3D triangle, how can I calculate which segment of the line (if any) is within a given distance d from the triangle?
Considerations:
Will ...
2
votes
2
answers
98
views
To find the length of the shortest path that begins at $(-1,1)$, touches the x-axis and then ends at a point on the parabola $(x-y)^2 = 2(x + y −4)$:
This is a question of a parabola but to solve it I need the coordinates of the vertex and focus which at present I'm unable to deduce in this form. I have learnt some standard form such as $ y^2=4ax$ ...
0
votes
1
answer
86
views
Intuitive explanation of perspective vs. projection in geometry
I was wondering if someone could please explain the intuition of the distinction between perspective and projection in geometry. Chapter 8 of John Stillwell's Mathematics and its History begins by ...
- 557
1
vote
5
answers
171
views
A Pythagoras-like conjecture using complex numbers
A physical experiment with weights, pulleys and a string led me to the following conjecture.
Let $a$ and $b$ be complex numbers having $\mathrm{Re}(a)<0$, $\mathrm{Re}(b)>0$ and $\mathrm{Im}(a)=\...
- 1,923
1
vote
1
answer
92
views
Area of an inscribed triangle
How to algebraically demonstrate that the statement below is false? A triangle always occupies more than a quarter of the area of the circle it circumscribes.
Visually, it is easy to see that the ...
- 5,971
1
vote
2
answers
104
views
Find the ratio of the areas of the circular regions below
Calculate the ratio of the areas of the circular regions below, knowing that the measurement of the ATB arc plus the measurement of the APB arc equals 450 degrees. (answer:$\frac{1}{2}$)
I try
$\...
- 5,971
2
votes
2
answers
221
views
Why is family of planes not working here?
Came across a question where it said $L_1$ is the line of intersection of the planes $2x−2y+3z−2=0$, $x−y+z+1=0$ and $L_2$ is the line of intersection of the planes $x+2y−z−3=0$, $3x−y+2z−1=0$.
I ...
- 45
12
votes
0
answers
295
views
Repeatedly flipping a three-segment wire: $2,\,\sqrt2,\,\frac{3+\sqrt5-\sqrt2\sqrt{1+\sqrt5}}2,\,\ldots\to\frac43?$
I have a three-segment wire on the plane with joints at $(0,1)$, $(0,0)$, $(r-1,0)$, and $(r,0)$, where $r>1$.
When $r=r_1=2$, I can perform two successive flips on segments of the wire to turn it $...
- 121
4
votes
1
answer
57
views
Permutations Given by Rotating Points in the Plane
Consider a set $X=\{p_1,p_2,p_3,...,p_n\}$ points in $\mathbb{R}^2$. Now consider the sets $R(X)$ for $R$ rotation matrices. For all but finitely many $R$, projecting $R(X)$ onto the $x$-axis gives an ...
- 2,286
2
votes
1
answer
70
views
Find the value of angle $x$ in the polygon below
The graph shows two regular polygons. If $O$ is the center of one of them, calculate $x$ (Answer: $82,5^{\circ})$
I found some Angles and tried to use the $\triangle DTS$ but I couldn't demonstrate ...
- 5,971
5
votes
3
answers
693
views
2 lines 1 plane question in 3 dimension
The question I need help with is
line $D: x = {y-2\over -1} = z$
line $D': {x-2\over2} = {y-3\over1} = {z+5\over-1}$
Find plane $(α)$ for $(α)$ containing $D$ and the angle between $(α)$ and $D' $ is ...
0
votes
1
answer
20
views
Are all convex hexagonal space tilings either double lattices or triple lattices?
Let $H$ stand for a convex hexagon with the following property:
It is possible to tile the space with $H$ using only translations and rotations.
There trivially are $H$ that tile the space with a ...
- 303
0
votes
0
answers
28
views
Percentage of acute triangles [duplicate]
There are 100 dots in a surface. A math lover draw all the triangles possible such that the vertices of the triangle will be those dots. X is the maximum percentage of acute triangles in those ...
- 11
0
votes
1
answer
41
views
Find a parametrization for the plane $E_1$ and a describing linear equation for the parametric representation of the plane $E_2$
a) Find a parametrization for the plane
$$E_1 = \\{ (x_1, x_2, x_3) \in \mathbb{R}^3 | 3x_1 - 2x_2 + x_3 = -1 \\} $$
b) Give a describing linear equation for the parametric representation
$$E_2 = (1, ...
- 1,141
1
vote
1
answer
369
views
Hess' Proof of Fuss' Formula
I am trying to understand the short proof of Fuss' Formula in the paper "Bicentric Quadrilaterals through Inversion" by Albrecht Hess which is available here:
https://forumgeom.fau.edu/...
- 1,430
1
vote
1
answer
232
views
Prove that if the normals of two hyperplanes are orthogonal, the vectors on the hyperplanes are orthogonal
For two planes:
$(\vec{p} - \vec{p_0})^T \vec{n_p} = 0$
$(\vec{q} - \vec{q_0})^T \vec{n_q} = 0$
Prove if the normals are orthogonal, i.e. $\vec{n_p}^T \vec{n_0} = 0$, then all vectors (except ...
- 679
3
votes
2
answers
211
views
Find the volume and total surface area of the solid formed
Let the planes be defined by $$|x|+|y|+|z|=1$$
Find the volume of the solid enclosed and the total surface area of the solid thus generated.
I am not able visualise the solid. What will it be$?$ I ...
2
votes
3
answers
164
views
A generalization of Pythagoras'
Consider a quadrilateral inscribed in a semicircle of diameter $d$, as in the picture below, then $$d^2 = a^2 + b^2 + c^2 + \frac{2 a b c}{d}$$
Notes:
If one of the $a$, $b$, $c$ equals $0$, we get ...
- 54k
1
vote
1
answer
26
views
Are the tangent lines at the farthest-separated points on a closed plane curve always parallel?
Suppose you have a closed differentiable plane curve. Are the tangent lines to the curve at the most distant points on the curve always parallel? What if we assume that the curve is convex?
I don't ...
- 5,989
0
votes
2
answers
172
views
Necessary condition for the product of two rotations of the plane to be a rotation
I know that any rotation of $\mathbb{R}^2$ can be expressed as the product of two reflections in non-parallel lines, and hence the product of two rotations can be written as $R_{L_4}R_{L_3}R_{L_2}R_{...
- 557
1
vote
2
answers
56
views
Distance of two parametrized lines
Consider the four points $A = (2, 4, 0), B = (3, 1, 1), C = (1, 1, 3), D = (0, 5, 1)$. Find the distance between the lines $(AB)$ and $(CD)$, i.e. the distance between the closest points on these two ...
- 93
2
votes
0
answers
107
views
The identity theorem for harmonic functions
$\textbf{Background for question}$:
Let $L_{k} = \{z \in \mathbb{C} : \arg(z) = \frac{\pi k}{n} \}$, and set $\Phi$ equal to the group of compositions of reflections in the lines $L_{k}, k=1,...,2n$.
...
- 2,060
0
votes
1
answer
68
views
direction vector of line
Could someone explain in simple words (maybe with a drawing) why the direction vector of
$$
\alpha x+\beta y+\gamma=0 $$
is equal to
$$
(\beta,-\alpha)
$$
- 61
1
vote
0
answers
18
views
Is the impression of an ideal boundary point (=end) the union of the impressions of the prime ends of the circle of prime ends associated to this end?
Let S be a compact orientable surface and U an open connected subset of S with finitely many ideal boundary points (or ends). U has a prime ends compactification which is a surface with boundary (...
1
vote
1
answer
79
views
Distance between two planes parallel to two lines
Consider the four points A : (2, 4, 0), B : (3, 1, 1), C : (1, 1, 3), D : (0, 5, 1). Find the
distance between the lines (AB) and (CD), i.e. the distance between the closest points on
these two lines, ...
- 93
2
votes
1
answer
128
views
Determine the value of $x+y$ in the triangle below
In figure , if the measure of angle $BQP$ measures $37$ degrees, determine the value of $x+y$.(Answer:$14-4\sqrt6$)
I try:
$\triangle PBQ:(3k-4k-5k) \implies 3k+4k = 2r+5k \implies 2k = 2.2 \...
- 5,971
2
votes
2
answers
81
views
Find the value of the segment parallel to the side of the triangle by an interior point [closed]
Let P be a point inside a triangle of sides a, b, and c through which they are drawn parallel to the sides of the triangle. If the parallel segments between the sides of the triangle have the same ...
- 5,971
0
votes
0
answers
15
views
Angle within a rectangular-based prism
I've tried to find solutions to similar problems to see how this question should be solved but I didn't have much luck.
A rectangular-based prism is shown below. I need to determine the size of angle ...
- 1
1
vote
1
answer
58
views
By how much would the length of the solar day change if Earth's rotation were suddenly to reverse direction?
The question in the title has to some extent been answered here:
https://worldbuilding.stackexchange.com/questions/79619/does-earths-direction-of-rotation-affect-day-length?newreg=...
- 27
0
votes
2
answers
109
views
A geometry problem: prove $AB=BF$. [closed]
There is a question aksed by my friend, who is a teacher of a milde school. But I failed to give an answer to him. The problem is
In the right triangle $\Delta ABC$, the angle $\angle BAC=\frac{\pi}{...
- 1,567
0
votes
1
answer
60
views
Find the measure of the $PQ$ segment in the regular octagon below
We have a regular octagon $ABCDEFGH$ with sides of equal length $\sqrt{2-\sqrt{2}}$, taking vertices $A, C$, and $E$ as centers, arcs of radii $AG, CA$, and $EG$ are drawn respectively, which ...
- 5,971
0
votes
1
answer
69
views
Maximum path length in a path-connected domain with smooth boundary
I have an open, bounded, connected, and simply connected domain $\Omega\subset\mathbb{R}^2$ with smooth boundary. I'd like to prove that there is a constant $M$ so that any two points in the domain ...
- 319
2
votes
2
answers
128
views
Determine the value, in degrees, of x in the quadrilateral below
In the figure below, $AB = BC$. Determine the value, in degrees, of x ($Answer:10^o$)
The solution I found was
$3x = x+20 \implies x = 10^o$
How to prove that the $\angle DBE=20^o$?
- 5,971
4
votes
0
answers
97
views
Hexagon to Rectangle dissection: 3 pieces minimal?
A hexagon can be divided into 3 pieces to make a rectangle.
Can we prove 3 pieces is minimal?
For a equilateral triangle to square dissection, it's thought that 4 pieces is minimal. We can prove that ...
- 20.7k
3
votes
1
answer
129
views
Express corners in a triangle from two angles and angle and polar coordinates of circumcenter
I have some lecture notes where the following is claimed:
Suppose that three points $0, p_1, p_2$ constitute a triangle $T$ in $\mathbb{R}^2$. Suppose the angle between $p_1$ and $p_2$ is $\theta$ and ...
- 1,031
3
votes
1
answer
97
views
Does every closed curve admit a line that intersects it at only one point?
Have a very basic question on closed plane curves, from this Wikipedia math reference desk conversation, that I can't seem to come up with any results for. In particular:
If $C : [0, 1] \rightarrow \...
- 67
0
votes
0
answers
43
views
Can a vertex lie on an edge in a planar graph?
I am wondering if a vertex can lie on an edge in a planar graph- I am not sure if an edge of this vertex is regarded as crossing the edge on which the vertex lies. I have two questions here:
Is the ...
- 2,145
0
votes
0
answers
34
views
Are the edges of a planar graph part of its faces? (Graph Theory)
The definition of face I have learned for planar graphs is "a region where any 2 points in it not on $G$ can be connected by a line which doesn't intersect any of the edges of $G$". I am ...
- 2,145
1
vote
1
answer
58
views
How is the face for a tree graph bounded by any sides at all? (Graph theory) [closed]
I have learnt that every face in a planar graph has sides, and that sides are edges which bound the face clockwise. I am very confused about a few things regarding sides:
I am not seeing how the ...
- 2,145
0
votes
0
answers
24
views
Connected components of image of non-degenerate boundary component
This is a follow up to my previous question, asked and answered here: Connected components of conformal image of boundary
I omitted this by accident from the last question, so I have created this ...
- 2,060
3
votes
1
answer
77
views
Connected components of conformal image of boundary
Let $f : G \rightarrow \mathbb{D}$ be a biholomorphism (a holomorphic map with holomorphic inverse), and suppose $G$ is a bounded open subset of the plane. Let $C$ be a compact connected subset of the ...
- 2,060
0
votes
0
answers
34
views
If 3 colours are used for a plane there exists a length 1 segment with edges of same colour.
A secondary school problem:
Prove that for any colouring of a plane with three colours there exists a segment of length 1 with edges of same colour.
My attempt to prove. Take any point A on the plane ...
- 807