Questions tagged [euclidean-geometry]

Geometry assuming the parallel postulate: in a plane, given a line and a point not on that line, there is exactly one line parallel to the given line through the given point.

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Are the only quadrilaterals satisfying this symmetric relation rectangles?

$\newcommand{\S}{\mathbb{S}^1}$ $\newcommand{\la}{\lambda}$ While solving an optimization problem, I reached the following question: Let $x_1,x_2,x_3,x_4 \in \S$ be four distinct points on the unit ...
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Find the length of CX

I found this question on Twitter My Attempt: I marked the circumcenter of the circle as O and drew radii $OA, OB, OC$ and with some angle chasing, I found the angles marked in the diagram attached ...
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$k$-dimensional subspace in $\mathbb R^n$ such that the orthogonal projection of standard basis $e_1, \dots, e_n$ have the same length.

I wonder how to construct or at least prove the existence of a $k$-dimensional subspace $V_k$ in $\mathbb R^n$ such that the orthogonal projection of canonical basis $e_1, \dots, e_n$ onto $V_k$ have ...
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On the possible cardinalities of Sylvester lines of sets of points in the real plane.

Let $n$ be a natural number greater than or equal to $3$. Also, let $S$ be a set of $n$ points in the real plane $\mathbb{R}^2$, such that there is no line that passes through all points of $S$. I ...
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The upper bound for the radius of the $k$-dimensional balls contained in an $n$ dimensional unit hypercube can be attained

It is shown in this thread that $\dfrac{1}{2}\sqrt{\dfrac{n}{k}}$ is an upper bound for the radius of $k$-dimensional balls that can be contained in an $n$ dimensional unit hypercube. But I have ...
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If $a$ and $b$ both closer to $x$ than $y$, prove that all points on line between $a$ and $b$ are closer to $x$ than $y$

For points $\mathbf{a}, \mathbf{b}, \mathbf{x}, \mathbf{y} \in \mathbb{R}^d$ for some natural $d \ge 1$. I want to prove that if (Euclidean distance) $||\mathbf{x} - \mathbf{a}|| \le ||\mathbf{y} - \...
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How many points $M\in(S)$ are there such that tangent plane $(P_M)$ intersects $Ox$ and $Oy$ at positive integer points and $\angle{AMB}=90^\circ$?

In a three-dimensional Cartesian coordinates system $Oxyz$, consider sphere $(S)\colon (x - 2)^2 + (y - 3)^2 + (z - 1)^2 = 1$. How many points $M$ which lie on $(S)$ are there such that the tangent ...
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What should I do if I have proved Euclid's fifth Postulate? [closed]

I have found a proof from Euclid's fifth Postulate and Corresponding Angle axiom in Euclidean Geometry. What should I do now?
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How to find $\frac{AC}{CE}-\frac{BD}{DF}$ given $\frac{AC}{AE}+\frac{DF}{BF}=1$?

The problem is as follows: Let $\textrm{A, B, C, D, E and F}$ collinear points and consecutive. It is known that, $$\frac{AC}{AE}+\frac{DF}{BF}=1$$ Find the value of, $$\frac{AC}{CE}-\frac{BD}{DF}$$ ...
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Showing that a curve from vertex $A$ of $\triangle ABC$ to side $BC$ intersects a curve from $C$ to side $AB$ at least once

Suppose that we have a triangle $\triangle ABC$, and two continuous curves inside it: one starts at the vertex A and ends on the side $\overline{BC}$, and the other one starts at the vertex $C$ and ...
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Prove that the sum of areas of triangles $AOH$ and $BOH$ equals the area of triangle $COH$.

Let $O$ be the circumcenter and $H$ the orthocenter of an acute triangle $ABC$. Prove that the area of one of the triangles $AOH, BOH,$ and $COH$ is equal to the sum of the areas of the other two. In ...
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Embeddings between Hamming and Euclidean space?

Recall the notion of Lipschitz function. Let $(M, d_M)$, $(N, d_N)$ be metric spaces, and let $\alpha > 0$. We say a function $f : M\to N$ is $\alpha$-Lipschitz continuous if $$\forall x,y \in M: ...
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Find the area ratio of the triangular regions $ATK$ and $LKS$ .

For reference: If, $T$ and $K$ are points of tangency, measure of the $\overset{\LARGE{\frown}}{AB}$ is twice the measure of the $\angle ASL$, $BS=3$, $KS=1$ and $\frac{TB}{4} =\frac{AT}{3}$ ....
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Finding distance from a point to a set

What is $d_A(p)$, where $A := \{(x,y) \in R^2 : x^2 + y^2 = 1\} ?$ Find an explicit expression. Also $p=(a, b) \in R^2$ This is a question from the book: Topology of metric spaces by S. Kumersan. I ...
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Find the area of ​the shaded region.

Given, $MT=2$ and $AC=8$. Calculate the area of ​​the shaded region. $\triangle BMT_(notable)\implies (a, 2a, a\sqrt5)\\ \therefore MT = 2, BT=4\\ \triangle ABF \sim \triangle MBT \implies k = \frac{...
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Area of a special triangle in a quadrilateral

As is the case with triangles, I am sure that there is an entire ocean of not-so-well-known theorems about Euclidean quadrilaterals. In particular, I am interested in the following problem and suspect ...
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2 votes
1 answer
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Is there a function $f$ from reals to reals such that every non-vertical line intersects $f$ infinitely many times?

Does there exist a function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for every non-vertical line $L$ in $\mathbb{R}^2$, $L$ intersects the graph of $f$ infinitely many times?
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Is there a subset of the real plane such that every line intersects that subset exactly once?

Consider the real plane $\mathbb{R}^2$. Does there exist a subset $S$ of the real plane, such that every line $l$ in $\mathbb{R}^2$ intersects $S$ at exactly one point?
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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: $OA =R\\ \triangle OAZ(equilateral)\implies S\triangle OAZ ...
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properties of symmetric difference of two set$|\Omega_1 \cap \Omega_2| |x_{\Omega_1} - x_{\Omega_2}| \le C(R) |\Omega_1 \Delta \Omega_2|$

how to show the following properties are holds in $\mathbb{R}^n$? for two bounded set $\Omega_1$, $\Omega_2 \subset \mathbb{R}^n$ and $\Omega_1$, $\Omega_2 \subset B_R$, $B_R$ is the Ball radius $R$ ...
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What is an upper bound on the diameter of a convex polytope?

Given a convex polytope defined by $Ax \le b$, with $V = \{ x_1, \ldots, x_n \}$ vertices, I would like to find the maximal distance $\max_{i,j} || x_i - y_i||_2$ as a function of $A$ and $b$ (some ...
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Proving an angle to be 90

$H$ is the orthocenter of acute triangle $A B C$. Let $\omega$ be the circumcircle of $B H C$ with center $O^{\prime}. \Omega$ is the nine-point circle of $A B C . X$ is an arbitrary point on arc $B H ...
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1 vote
1 answer
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What is the cardinality of the set of shapes?

Consider the Euclidean plane $\mathbb{R}^2$. A figure is a subset of the Euclidean plane. Two figures $S$ and $T$ are said to have the same shape iff there is a composition of an isometry and a ...
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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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What does it mean for a set of points to be affinely independent in the context of range spaces?

For some background, there are many range spaces with finite VC-dimension that arise naturally in discrete and computational geometry. One example is the set of all points in d-dimensional Euclidean ...
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i'm looking at Euclid's elements i see all his propositions on ratio and proportion but im trying to figure out how it relates to our modern algebra? [closed]

could anyone point me in the direction of a resource for this, or possibly give an explanation? I'm basically wondering how the rules of algebra could be developed into what they are now.
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Getting better at Geometry - soft question

I am a high school student in the UK (year 12 UK, grade 11 US. I am very interested in maths and so have been doing some STEP papers (2 and 3) in my spare time. I have become reasonably proficient at ...
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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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3 votes
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Suppose that $MN=5\sqrt{2}|AM-BN|$ reaches its maximum value when $A=(-4;7;3),B=(4;4;5),M=(a;b;c)$ and $N=(d;e;f)$, determine the value of $a+d$.

In a three-dimensional Cartesian coordinates system $Oxyz$, consider $\vec{a} = (1; -1; 0)$ and two points $A(-4; 7; 3), B(4; 4; 5)$. Suppose $M, N$ are two points in plane $(Oxy)$ where $\...
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1 vote
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Calculating the average distance between infinite random point pairs within a polygon

I need to come up with a mean figure for the distance between any two points within an irregular polygon. Is there a name for what I'm looking for? All distances would be as-the-crow-flies. Thanks ...
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1 answer
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How can I prove that the perimeter is at most 60?

Problem: Let $\Delta$ be a triangle in the plane. Let $P$ be the perimeter of the triangle and $A$ be the area. Let $a,b,c$ be the length of the sides and suppose they are positive integers. Suppose ...
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3 votes
1 answer
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David Hilbert's Foundations of Geometry, section 9, Compatibility of the Axioms

I am reading David Hilbert's Foundations of Geometry. In section 9, where he shows the "Compatibility of the Axioms" he begins with the following: Let us consider the domain $\Omega$ ...
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3 votes
1 answer
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Proving a linear transformation that preserves the inner product is an isometry

I am currently working on a problem to prove the following statement: Suppose $T: \mathbb{R}^n \to \mathbb{R}^n$ is a linear transformation. Prove that $T$ is an isometry if and only if $T(v) \cdot T(...
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5 votes
1 answer
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How to define a (linear, invertible) mapping from a square to a triangle

This problem has come up when analyzing one type of HSV color selector: Note: h,s,v are in 0...1 range. For the given hue (i.e. red) on the three vertices of the triangle we have: W: white, when s=0,...
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2 votes
2 answers
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Prove the independence of a certain segment in a special triangle

Let $\triangle ABC$ be a triangle with obtuse angle $\angle A$ and $\overline{AB} = 1$. Also, let $\angle C = \gamma$ and $\angle B = 2\gamma$. If $E$ and $F$ are intersection points of perpendicular ...
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0 answers
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Trying to prove the inner ball condition for $C^2$ domain

I am trying to prove the inner ball condition for a $C^2$ domain $\Omega$. Let $a\in \partial \Omega$ since $\Omega$ is $C^2$ there are $r>0$ and $f:\Bbb R^{d-1}\to\Bbb R$ a $C^2$-function such ...
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Modern axiomatically rigorous version of Euclid's Elements

I have been wanting to read Euclid's Elements (Oliver Byrne's version) for a few months, but I have recently learned that a number of the proofs in Euclids Elements are not very rigorous, and that the ...
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1 answer
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Can I assume there is homothety in this situation?

I have found two similar triangles,the triangles $\triangle ABC$ and $\triangle A_1B_1C_1$. The lines connecting their respective vertices intersect at exactly one point, in other words concurrent. ...
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3 votes
2 answers
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Prove that $AG \parallel BE$

As shown in the picture, $E$ is a point outside the square $ABCD$ Connect $BE,CE,DE$, point $F$ is on line $DE$ Connect $AF$ to intersect line $DB$ at point $G$ Suppose that $DE=DB, CE=CF, AG=EB$ ...
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2 answers
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Determine the radius of circle $(C)$ which is less than those of circles $(C_1), (C_2), (C_3)$ that are tangent to $(C)$ and to one another.

Consider four circles $(C_1), (C_2), (C_3)$ and $(C_4)$ which all lie on a sphere of radius $2$ and are tangent to one another. The radius of circle $(C)$ is less than those of circles $(C_1), (C_2), (...
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Property about complete cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral inscribed in a circle of center $O$ and let $AB\cap DC=\{E\}$, $AD\cap BC=\{F\}$ and $AC\cap BD=\{X\}$. Let $OE\cap FX=\{G\}$ and prove that quadrilaterals $AOGB$ ...
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2 answers
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Why are these three intersection points collinear?

This is what I found several years ago when I was in middle school: Suppose we have a circle on a plane and arbitrarily choose four different points on the circle, say $P,A,B,C$. Then draw three ...
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Proving $T$ is an isometry iff it preserves the inner product

I'm working on an exercise in the text and am proving the following statement: Suppose $T: \mathbb{R}^{n} \to \mathbb{R}^{n}$ is a linear transform. Prove that T is an isometry if and only if $T(v) \...
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4 votes
3 answers
143 views

Need help with Euclidean geometry

Here's the problem: Consider triangle ABC inscribed in circle C(O, r). The angle bisector of ∠A (resp. ∠B, ∠C) intersects the circle C(O, r) in the points A and A' (resp. B and B', C and C'). Prove ...
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2 votes
2 answers
83 views

Showing that a quadrilateral is a square

I am trying to prove the following: "Consider a square ABCD. Draw an external line r through D and call H and K the projections of A and B on r respectively, and with R the projection of A on BK....
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Can a non-degenerate polygon with all sides equal have unequal angles?

I have always been hearing that a regular polygon is a polygon with equal sides and equal angles, but I never considered the fact that it may be possible for a polygon with all sides equal but unequal ...
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0 answers
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Example of $C^{1,\alpha}$ domain not satisfying the interior sphere condition

I would like to prove there exists $C^{1,\alpha}$ with $0<\alpha<1$ not satisfying the interior Sphere condition. I consider $\Omega=\{(𝑥,𝑦)\in \Bbb R^2:𝑦>|𝑥|^{1+\alpha}\},$ with $0<\...
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3 answers
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How can I get direction vector from normal vector in 3d?

Lets say I have normal vector $(x,y,z)$ how can I get direction vector from it? In $2d$ is simple as far as I know just change $\pm$ sign of one of two values but how it works in $3d$? Thank you.
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1 vote
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If $f \in I(\mathbb{R}^2)$ and $T_c$ is a translation, then ${T_c}^{-1} \cdot f \cdot T_c$ is the same kind of isometry as $f$.

I want to know if this statement is true. If $f \in I(\mathbb{R}^2)$ and $T_c$ is a translation, then ${T_c}^{-1} \cdot f \cdot T_c$ is the same kind of isometry as $f$. This question arises from a ...
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6 votes
0 answers
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Which objects can be Minkowski halved?

The Minkowski Sum of two subsets $A,B \subset \mathbb{R}^n$ is $$A \oplus B = \{a + b | a \in A, b \in B\}$$ For a given $A$, is there some condition that tells me when I can find a $B$ such that $A = ...
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