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.
2,112 questions
8
votes
1
answer
457
views
A subset of a rectangle that 'blocks' every curve that goes from right to left must connect upper and lower sides
I'm trying to find a proof for the following assetion: Given a rectangular region $R$ and a subset $A$ of $R$, if every curve that starts at the left side of $R$ and ends at the right side intersects $...
- 333
0
votes
1
answer
57
views
Collinearity of three points in a square with perpendicular and parallel constructions
Given a square $ABCD$ with $E$ an interior point on side $CD$ (not at the endpoints).
Construction:
From vertex $D$, draw ray $Dx$ perpendicular to $AE$, intersecting side $BC$ at point $H$
From ...
6
votes
2
answers
263
views
Is there a more intuitive way to prove that the locus of this point is an arc of a circle?
Given that the position of line segment $AB$ is fixed, there is a fixed line above it (the blue line in the figure), and there is a moving point $P$ on the line. Connecting $PA$ and $PB$, if $Q$ is on ...
- 281
7
votes
4
answers
251
views
Find the measure of $∠OTS$ in a circle
I need help with the following Geometry problem.
In a circle with diameter $AB$ and center $O$, chords $PQ$ and $QC$ are drawn that intersect $AB$ at $M$ and $N$, respectively. If the extensions of $...
- 81
-1
votes
1
answer
102
views
Need help solving "sets + geometry" problem
Problem
Find the smallest number $n$ such that there exist sets $S_1,...,S_n$ satisfying:
For all integers $1 \leq i < j \leq n,$ we have $S_i \cap S_j = \emptyset.$
$S_1 \cup S_2 \cup \cdots \...
2
votes
3
answers
147
views
Determine the maximum integer value in the figure
From the figure determine the maximum integer value of $PN$.
$PE=EN=EC$
$AB=4:AD=5$
(Answer: $3$)
I determined some angles but couldn't develop it.
$\angle ADP = 90^o -\beta\\
\angle ABD =180^o -2\...
- 7,873
0
votes
2
answers
96
views
Need solution expalnation of this plane geometry problem
This is the problem:
Let $ABCD$ be an isosceles trapezoid with $AB \parallel CD$ with $AB = 5$, $CD = 15$, and $\angle ADC = 60^\circ$. If $E$ is the intersection of $\overline{AC}$ and $\overline{BD}$...
- 1,916
0
votes
0
answers
62
views
A tangent sphere is constructible in $3$D, if the corresponding tangent circle is constructible in $2$D?
I came across the following problem:
As shown in the right figure, four tangent spheres are placed inside a cube. What is the radius of the fifth blue sphere that is tangent to all these spheres?
...
- 8,489
0
votes
0
answers
153
views
Calculate the sum of the maximum and minimum integer values that θ can assume.
In a triangle $ABC$, another triangle $ACD$ is constructed externally. Then, $BD$ is drawn, which intersects the side $AC$ at point $O$ such that:
$AB=CD=OD$
$m∠ADB=20^o $
$m∠BAC=m∠CAD=θ$
Calculate ...
- 7,873
0
votes
1
answer
84
views
Pair of planes of a pair of lines.
Edit: Apologies for the confusing wording, I'm a first time poster here.
Question
I was attempting this question from the MIT opencourseware Multivariable Calculus course.
"Consider the four ...
- 1
1
vote
0
answers
69
views
Existence of four concyclic points on the graph of a real function under scaling of abscissa
Suppose $f:\mathbb R \to \mathbb R$ is a smooth function.
Fix four distinct reals $u_1<u_2<u_3<u_4$. For each $x_0\in\mathbb R,\lambda\in\mathbb R-\{0\}$, define four points
$$
\bigl(x_0+\...
- 184
3
votes
1
answer
82
views
A property of spiral closed curves
I have a question related to a simple property of a spiral closed curve, by which I mean figures of the following kind:
I want to somehow prove that this kind of a closed curve satisfies both the ...
- 159
3
votes
1
answer
186
views
Find the angle $x$ in this triangle.
Find the angle $x$ in $\triangle ABC$ if $AB=AC,AP=QL$ and $PQ=LC.$
(Answer: $26.5^\circ$)
I have a solution involving trigonometry, but I need a geometric solution.
Use $\tan(2θ)$ and $\tanθ$ we ...
- 7,873
0
votes
1
answer
66
views
Calculate section MN between the centers of internal circles under the conditions below
$4$ circles of the same radius $r$ are internally tangent to a bigger circle centered at $O$ with radius $R$. Additionally the circles centered at $M,S$ or $N,T$ are externally tangent.
Given $MN\...
- 1
7
votes
7
answers
335
views
Under the conditions given below, calculate the angle $ABP$ of the triangle $ABC$.
In $\triangle ABC$ a point $P$ is located external to side $AC$ such that $BC = CP = PA.$ $\space$ If $m∠BAC = 75^\circ \text{ and } m∠BCP = 90^\circ$ then find $m∠ABP. \space$ (The answer is supposed ...
- 7,873
6
votes
8
answers
316
views
Rotating/tilting conics
There is a formula for the inclination/tilt $\theta$ of a conic given by a level curve of the $Ax^2+Bxy+Cy^2$, namely: $\cot 2\theta=\frac{A-C}{B}$. This can be derived by a straightforward ...
- 49.2k
5
votes
1
answer
132
views
Can a planar region's shape be determined by the areas enclosed by certain chords?
Consider a compact, simply-connected planar region $A\subset\mathbb{R}^2$ with bounded perimeter whose boundary is endowed with an isometric parameterization $r:S^1\rightarrow\partial A.$ For each $0&...
3
votes
1
answer
93
views
Construct a polygon $H$ with a point $P$ outside it such that no side of $H$ is completely visible from $P$
Question: Construct a polygon $H$ with a point $P$ outside it such that no sides of $H$ is completely visible from $P$.
I find this question a little bit tricky. The case where $P$ is inside the ...
0
votes
1
answer
66
views
Lines divide plane into areas with at least three sides
Here is an exercise in Polynomial Methods in Combinatorics.
Suppose that $\mathfrak L$ is a finite set of lines in $\mathbb{RP}^2$. We say the lines of $\mathfrak L$ are concurrent if there is a ...
- 845
2
votes
1
answer
49
views
Counterexample to transitivity of equidistance relation of congruent connected sets in $\mathbb{R}^2$
Let $(X, \operatorname{dist})$ be a metric space. Two subsets $P, Q \subseteq X$ are said to be parallel if there exists $d \geq 0$ such that
$$
\operatorname{dist}(p, Q) := \inf_{q \in Q} \...
- 6,087
2
votes
1
answer
246
views
Question about equilateral triangle that packed into a square
I saw this question.
Looking at the picture in this question, I have another question. If an equilateral triangle is tightly packed into a square, and one vertex of the triangle share the vertex of ...
- 867
5
votes
0
answers
148
views
How many times a 2-dimensional shape can be folded? [closed]
I was thinking about this problem:
How many times can I fold a 2-dimensional shape so that the two resulting half can be overlapped?
Although it seems not so difficult, I would like to adopt a ...
- 399
1
vote
0
answers
112
views
What math subjects are relevant for someone wanting to learn how and why plane and solid geometry, projective geometry work in the context of art?
everyone! I am hoping to get some direction and book recommendations. I am an artist and have been learning from an art teacher a little about the role that geometry played for the Old Masters and the ...
- 11
0
votes
1
answer
55
views
How do I calculate the signed distance between a point and the boundary of a shape that is composed of multiple short parametric curves?
As part of a personal project, I have divided the Cartesian plane into patches bounded by parametric curves. Given the set of curves that surround any given patch, I want to find a 2D function that ...
- 2,153
2
votes
1
answer
100
views
Are all Euclidean Triangle Groups also wallpaper groups?
I've been working through D.L. Johnsons book Symmetries and am currently in the chapter about triangle groups, specifically the part about triangles in the real plane, where there are only three cases....
- 59
3
votes
1
answer
80
views
“Formula for Counting Regions Created by Parallel and Perpendicular Lines in a Box
I was playing with a simple geometry idea and noticed a pattern. If I draw n parallel horizontal lines inside a box, they divide the box into n + 1 regions. ( like put one line down a box and you end ...
- 31
1
vote
2
answers
195
views
intersection between plane and circle arc
I have a 3D scene which is discretized into a set of plane. When interacting with the scene, I project circle arcs into the 3D space.
I know all the details of the planes (point location, orientation, ...
- 37
0
votes
1
answer
99
views
When cutting a square into two equal shapes: does the dividing line pass through the centre of the square? Does the shape have symmetry of order $2?$
Following on from my previous question,
Let $X\subset\mathbb{R}^2$ be a square including it's inside space,
not just the edges/boundary. Suppose that $Y$ is a (not necessarily straight)
line/curve/&...
- 25.2k
2
votes
1
answer
117
views
When cutting a polygon into two equal shapes: does the dividing line pass through the centre of mass? Does the polygon have symmetry of order $2?$
Let $X\subset\mathbb{R}^2$ be a polygon including it's inside space, not just the edges/boundary. Suppose that $Y$ is a straight line that cuts the polygon into two equal shapes, $X_1$ and $X_2:=X\...
- 25.2k
1
vote
2
answers
178
views
Calculate the area of the quadrilateral MNFE (shaded region).
$P$ and $Q$ are points of tangency. $I_1$ is the incenter of triangle $ABH$ with radius $r_1=3u$. $I_2$ is the incenter of triangle $HBC$ with radius $r_2=4u$.
Calculate the area of the quadrilateral ...
- 7,873
2
votes
1
answer
128
views
Calculate the area of the shaded region $BMP$ in a square $ABCD$
"If $ABCD$ is a square, calculate the area of the shaded region if $MN = 1m$
(Answer: $3m^2$)
I found the solution below online, but it was through analytical geometry. I would like to know if ...
- 7,873
2
votes
1
answer
219
views
Find area of the triangle formed by the centers of three circles.
Express the area of the shaded region (triangle $OIE$) using only the lengths of segments shown in the following picture.
I tried but I couldn't finish
$a$ and $b$ be the lengths of the arrows (...
- 7,873
2
votes
3
answers
509
views
Find the angle $x^\circ$ in the semicircle
Calculate the angle $x^\circ$ in the semicircle, if $\overline{PQ} = 2\,\overline{QN}$.
(Answer: $15^\circ$)
My Approach
Let $\overline{AP} = y,\; \overline{QN} = k, \;\overline{AO} = R$
$\angle OMB = ...
- 7,873
2
votes
3
answers
276
views
Find the value of alpha in the quadrilateral below.
Find $\alpha$
(Answer:$10^o$)
Anyone have any ideas? I calculated the angles, tried some auxiliary lines, but I can't find a relationship that gives alpha."
$\angle ABE = 90^o -2\alpha\\\angle ...
- 7,873
0
votes
0
answers
28
views
Calculating angular velocity on intermediate axis
I have a gyroscopic sensor and it can provide angular speeds in all 3 axes in degress / second. I want to calculate angular velocity on an axis exactly in the middle between X and Y.
I'm adding ...
- 101
2
votes
2
answers
128
views
Find $x^o$ in the quadrilateral $ABCD$ below"
Calculate $x^o$
(Answer:$\frac{37}{2}$)
I tried but I'm unable to finish.
Extend $CB$ to $N$ on $AD$
$\angle CAN = 180^o -2x \implies \angle ANC = x\\
\therefore \triangle ACN_{(isos)}\\
AH \perp NC \...
- 7,873
0
votes
0
answers
77
views
In the figure below with the indicated angles, calculate the value of angle $x$.
In the figure below, if $AB=2BC$, calculate $x^o $.
(Book's answe: $38^o$)
I was able to find this solution for the problem above, but I need a solution using auxiliary lines. Does anyone have one?
$...
- 7,873
0
votes
1
answer
45
views
If a quadrilateral $ABCD$ is cyclic, then $\angle ABD\cong \angle DCA$. But is the converse true?
I know that if a quadrilateral $ABCD$ is cyclic, then $\angle ABD\cong \angle DCA$. But is the converse true? That is, if in a quadrilateral $ABCD$ we can verify that $\angle ABD = \angle DCA$, can we ...
- 304
3
votes
2
answers
207
views
Geometrical interpretation of secondary solution in incircle and tangent circle problem
I am referring to GoGeometry problem 454. Here is a diagram of the problem with some notation
the remarkable identity that needs to be proven is $r=\sqrt{r_1r_2}+\sqrt{r_2r_3}+\sqrt{r_3r_1}$.
One ...
- 900
0
votes
1
answer
68
views
Find inner product plane geometry
In triangle ABC, $\overline{AB}=1$, D is the midpoint of $\overline{AB}$, E is on $\overline{AC}$, with $\overline{AE}:\overline{EC}=2:1$. If $\overline{BE}$ and $\overline{CD}$ intersect at P, ...
- 43
0
votes
2
answers
109
views
Calculate the perimeter of the quadrilateral formed by joining the midpoints of the quadrilateral $C1, C2, C3$ and $C4$.
"In a trapezoid ABCD, C1, C2, C3, and C4 are the centroids of the triangular regions ABD, ABC, BCD, and ACD. The perimeter of the quadrilateral formed by joining the midpoints of the sides of the ...
- 7,873
10
votes
3
answers
2k
views
How do you find the exact area enclosed by a closed curve?
For example, how do you calculate the area of the shape below or a general non-function that describes a solid?
$$\left((x+2)^2 + (y+2)^2 \right) \left[\left(\sin\left(\frac{3 \pi^2}{4} x\right) + \pi\...
- 379
0
votes
2
answers
77
views
What value can the perimeter of the quadrilateral described $ZUVW$ assume?
Given a convex quadrilateral $ABCD$, the points $M, N, P$, and $Q$ are taken on the sides, such that point $M$ is on side $\overline{AB}$, the point $N$ is on side $\overline{BC}$, point $P$ is on ...
- 7,873
0
votes
1
answer
135
views
In the quadrilateral ABCD,AM=MC=BN=ND=4; Calculate MN.
In the quadrilateral $ABCD,AM=MC=BN=ND=4$;
Calculate $MN$.
(Answer: 4)
Does anyone have a more geometric solution using auxiliary lines?
Let $x= OM$ and $y =ON$
$O$ is the intersection of the ...
- 7,873
3
votes
2
answers
188
views
Elliptic Constructibility of the Heptagon
I was reading this paper where at the end they add the remark:
By "elliptically constructable" they are referring to introducing the ability to draw ellipses using two fixed foci and a ...
- 1,849
0
votes
1
answer
63
views
Show construction of notable point does not depend on initial choices without using coordinates
In the Euclidean plane we are given a line $r$, a circle $c$ -with center at $C$- and a point $A$ like in the picture below (these are the elements drawn in black color):
Now consider the line $s$ ...
- 1,702
8
votes
2
answers
481
views
The geometry behind $\cos 48^{\circ}$ as decomposition of $\sin 18^{\circ}$
Consider a $\triangle ABC$ with $\angle A = 18^{\circ},~\angle B = 78^{\circ},~\angle C = 84^{\circ}$. The height $\overline{AD} \perp \overline{BC}$ splits the apex into $\angle DAC = 6^{\circ}$ and $...
- 7,315
1
vote
2
answers
473
views
In triangle ABC Calculate $\alpha$ if BC=AD [closed]
Calculate $\alpha$ if $BC=AD$
(Answer:$10^o $)
I have the resolution below but I would like a geometric resolution by auxiliary lines
Sine Theorem in $\triangle{BCD}$:
$$\frac{BC}{BD}=\frac{\sin(180−3\...
- 7,873
2
votes
0
answers
56
views
Convex shape B when translated in some direction decreases its intersection with convex shape A, will their intersection continue to decrease further? [duplicate]
Two convex shapes $A$ and $B$ intersect on plane. If $B$ is translated at arbitrarily small distance in vertical direction its intersection with $A$ decreases. Prove (or find counterexample) that ...
- 824
4
votes
1
answer
364
views
Relationship of distances between centers of circles
If: $O_1; O_2; O_3; O_4,$ they are also centers: $\overline{PO_1} = 8$m; $\overline{PO_2} = 4$m and $\overline{PO_4} = 12$m. Calculate $\overline{PO_3}.$
(Answer:$6$m)
I try but I can't get the ...
- 7,873