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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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Combination of 1981 glide reflections in $\mathbb{E}^2$ still a glide reflection?

I was wondering if the combination of 1981 glide reflections over different lines is still a glide reflection over a line in $\mathbb{E}^2$ (so every glide reflection can be over a different line). Or ...
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Existence of a triangle with sides partitioned in particular ratios by inscribed circle

Is there a triangle with sides that are partitioned into line segments of ratios $3:2$, $3:5$, and $10:9$ by the points of tangency of its inscribed circle? By Ceva's Theorem, if a triangle has sides ...
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Prove that the locus of the midpoint of a family of parallel chords of a circle is a diameter which is perpendicular to the given family of chords. [on hold]

Prove that the locus of the midpoint of a family of parallel chords of a circle is a diameter which is perpendicular to the given family of chords. My earlier question asked for a figure to help ...
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2answers
28 views

Show that the locus of midpoints of a family of parallel chords of a circle is a diameter perpendicular those chords.

Show that the locus of the midpoint of a family of parallel chords of a circle is a diameter which is perpendicular to the given family of chords. Please help me understand the question with a figure ...
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2answers
25 views

Finding the internal angles of a polygon inscribed in a heptagon

Consider this heptagon whose side measure is $1$ and prove $\angle FAB = \displaystyle{4\pi\over 7}$. I want to know the measure of the $\angle FAB$. My textbook states that it is given by $\frac{...
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A dummy but tricky optimization problem: how to link “projections” of different euclidian norms?

Ciao, I am working on the following optimization problem. Let's define: $$ \Pi_n(k) = || kw-v||_n^n $$ where $v, w \in \mathbb{R}^m$ are two fixed vectors and $|| \cdot ||_n$ is the usual n-norm. ...
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Intuition/Motivation - Distance in Euclidean Space

In modern mathematics, euclidean distance is defined using the Pythagorean Theorem, that is, by a formula such as $\sqrt{(x_1 - y_1)^2 + \cdots + (x_n - y_n)^2}$. A priori there is not a reason why ...
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43 views

If circles with centers $C_1$, $C_2$ and radii $R_1$, $R_2$ meet, then $R_1-R_2<|C_1C_2|<R_1+R_2$ [on hold]

When two circles of centre $C_1$ and $C_2$ and radius $R_1$ and $R_2$ cut, prove that $$R_1-R_2 < |C_1C_2|$$ $$R_1+R_2 > |C_1C_2|$$ I tried making a right triangle with sides $R_1$ and $...
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A line connecting an interior and an exterior point of a circle should intersect the circle at some point

How can someone prove in Euclidean geometry that the statement "A line connecting an interior and an exterior point of a circle should intersect the circle at some point" follows from the axioms ...
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Constructing a quadrilateral using the method of translation - Kiselev

Ending the fourteenth chapter of Planimetry is the following construction: Why is this a construction, when the quadrilateral in question is already given.
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How to find the smallest side of a triangle when the interior angles are unknown?

I am confused on how to find the answer for this problem. So far what I believe would apply is the triangle inequality but I'm not sure on how to use it. The figure $ABC$ is a triangle so as $BDC$. ...
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4answers
52 views

Congruent triangles in 3 tangent circle configuration

Two circles $\mathcal{C}_1$ and $\mathcal{C}_2$ of centers $O_1$ and $O_2$ are externally tangent at $I$ and internally tangent to a third circle $\mathcal{C}$ of center $O$ that is colinear with $O_1$...
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Proving $x_1y_1 +x_2y_2 + kx_1y_2 + kx_2y_1$ defines a scalar product in $\mathbb{R}^2$ if $|k|$ <1 [on hold]

Prove $x_1y_1 +x_2y_2 + kx_1y_2 + kx_2y_1$ defines a scalar product in $\mathbb{R}^2$ if $\left|k\right|<1$ Using definition that scalar product is a bilinear function which satisfies linearity, ...
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Steiner tree to minimise travelling distance: Building roads to connect a network of points

Suppose we have four points in a unit square, as described in the question here. We are tasked with building a network of roads that connect all the cities. The travelling distance (T) of this network ...
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1answer
78 views

Pythagoras-like equality in a problem [on hold]

Known information: $△ ABC$ is rectangular and isosceles $M$, $N$ belong to hypotenuse angle measurement $\angle MAN = 45^\circ$ Requirement: to demonstrate that $$BM^2 + CN^2 = MN^2$$ Is a ...
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A rotating polygonal line with increasing side length cannot end up where it started!

Consider a polygonal line $P_0P_1...P_n$ such that $\angle P_0P_1P_2=\angle P_1P_2P_3=...=\angle P_{n-2}P_{n-1}P_{n}$, all measured clockwise. If $P_0P_1>P_1P_2>...>P_{n-1}P_{n}$, $P_0$ and $...
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Curve with discrete self-similarity under proper similarity?

I've been playing around with the idea of curves that have a discrete self-similarity. What I mean by this is that we pick a similarity transformation $T$ in the Euclidean plane, and we look for a ...
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3answers
498 views

Ways of geometrical multiplication

There are at least five ways to multiply two natural numbers $a$ and $b$ given as integer points $A$ and $B$ on the number line by geometrical means. Two of them include counting, the others are ...
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1answer
25 views

Distance between a point and low-dimensional sphere

Is there a way to analytically calculate the distance between an arbitrary point $\mathbf{x}\in\mathbb{R}^n$ and a low-dimensional sphere embedded in $\mathbb{R}^n$, say one aligned with the axis ? ...
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Plane Geometry related to Circle

The internal bisector of angle A of triangle ABC meets the circumcircle in D. If DE and DF are the perpendiculars to AB and AC respectively from D. Prove that AE is arithematic mean of AB and AC .
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Are there always two circles that together surround or intersect all points in the following scenario?

Consider $N$ points in $\mathbb{R}^2$ and $\binom{N}{2}$ circles, one for each pair of points such that it intersects both. Is it always possible to pick two of these circles that together surround or ...
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Prove that $EF'$ passes through the circumcenter of $\triangle E'M'N'$.

$F$ is a point outside $\square ABCD$ such that $\widehat{CFD} = 135^\circ$. $FF' \perp CD$ at $F'$. $AC \cap BD = \{E\}$. $AF \cap BD = \{M\}, AF \cap CD = \{M'\}$ and $BF \cap AC = {N}, BF \cap DC = ...
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Finding Overlap of polygons in 3D space

I'm trying to find the amount of "overlap" between two (or more) polygons in a 3D space. The planes all have vector normals pointing in the same direction, so they are guaranteed to be parallel to ...
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3answers
49 views

How to find the angle in a protein which is inside of a triangle which appears inscribed in a circle?

I'm confused at which property or identity can be used to find the angle in a triangle when it looks inscribed in a circle but one of its sides doesn't appear to pass through the center. I'm also ...
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3answers
139 views
+50

High school challenge problem regarding perimeters of triangles

Can anyone help with Q25, above? I have tried applying the sine and cosine rules, arguments about similar triangles, and general diagram-chasing to no avail. I am somewhat confused by the condition ...
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Is there a relation between Hamming distance and Euclidean distance?

Is there a relation between Hamming distance and Euclidean distance? If components of vectors are $\in \{1,2\}$,for example. For $\{0,1\}$ everything is obvious.
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1answer
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How to construct a normal to side $AB$ of acute triangle so that it halves it area?

How to construct a normal to side $AB$ of acute triangle so that it halves it area? So we have $${c\cdot v\over 2} = 2{x\cdot v' \over 2} \implies {v\over v'} = {2x\over c}$$ From similar ...
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Borel measurable function on the Euclidean space $\pmb{R}^4$

I want to know if my function $f$ is Borel measurable function or not. To be clear, I use the terms which are introduced in $\\$https://dspace.mit.edu/bitstream/handle/1721.1/14254/22712180-MIT.pdf?...
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1answer
25 views

What does Equidimensional equation mean

What does the adjective equidimensional mean ? I encountered it while studying Cauchy-euler equation in differential equations with applications and historical notes by gf simmons : http://...
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1answer
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If a ray r emanating from an exterior point of triangle ABC intersects side AB at any point between A and B, then r also intersects side AC or side BC [duplicate]

Prove Proposition 3.9: If a ray r emanating from an exterior point of triangle $ABC$ intersects side $AB$ at any point between $A$ and $B$, then $r$ also intersects side $AC$ or side $BC$. Can ...
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4answers
260 views

Equilateral triangle on a concentric circle

Is my idea correct? 3 concentric circles of radius 1, 2 and 3 are given. An equilateral triangle is formed having its vertices lie on the side of the three concentric circles. What is the length of ...
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3answers
52 views

Some questions on the intersection of three cones.

I have three cones in $\mathbb{R}^3$, explicitly defined by the equations: $$ (x-\alpha_x)^2+(y-\alpha_y)^2=(z-r_1)^2 \,, \\ (x-\beta_x)^2+(y-\beta_y)^2=(z-r_2)^2 \,, \\ (x-\gamma_x)^2+(y-\gamma_y)^2=(...
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ABCD and AECF are two parallelograms and side EF is parallel to AD . suppose AF and DE met at X and BF AND CE AT Y . prove that XY is parallel to AB

I tried proving it by showing angles exy and eyx equal to edc and ecd respectively but I got no where . Is there any other approach I should consider
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Angle of a Star Inscribed in a Circle

I don't even know where to start on this: In the figure, point O is the center of the circle, points A, B, C, D and E all lie on the circle, and both segment AD and CE go through point O. Angle BEC ...
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Area of the intersection of two triangles.

Let $\triangle{ABC}$ be a triangle with $AB=5$, $BC=7$, and $CA=4$. Define $D$, $E$, and $F$, to be the midpoints of $AB$, $BC$, and $CA$ respectively. Let $G$ the intersection of the medians of $\...
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$\sum AB\left( \sin \widehat {CAP}+\sin \widehat {CBP}\right) \leq \sum AB$

In the triangle $ABC$ we consider the point P. Prove that $AB\left( \sin \widehat {CAP}+\sin\widehat {CBP}\right) +BC\left( \sin \widehat {ABP}+ \sin\widehat {ACP}\right) +CA\left( \sin \widehat {BCP}+...
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Existence of the inscribed hypersphere of a simplex

Letting $\textbf{T}=[u_0,...,u_n]$ be a $n$-simplex of $\mathbb{R}^n$, how does one prove the existence of the inscribed hypersphere ? Looking at the possible duplicates, people only seem to ask what ...
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What are the details of this step in a proof of the Banach-Tarski paradox?

In this exposition of the Banach-Tarski paradox by Terry Tao, Corollary 1.4 says, There exists a partition $S^2 = \Gamma_1 \uplus \dots \uplus \Gamma_8$ and rotation matrices $R_1, \dots, R_8 \in ...
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2answers
51 views

In a quadrilateral ABCD ,which is not a parallelogramm. On the rays AB,CB,CD,AD we put… [closed]

In a quadrilateral $ABCD$, which is not a parallelogram, on rays $AB$, $CB$, $CD$, $AD$ we put points $K$, $L$, $M$, $N$ such that $KL\parallel MN\parallel AC$ and $LM\parallel KN\parallel BD$. Prove ...
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Compute numerically the angle $x$ in the triangle without trigonometry

Be $\triangle CAB$ right in $A$ such that $AB=a$ and $\angle CBA = \alpha$. Extend $BC$ to $D$ such that $\angle CAD=2\alpha$ and $\angle ADC=x$. If $M$ is a point $\in BC$ such that $BM=MC$ and $MD=...
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Geometry high school math competition question

Let $ABC$ be an equilateral triangle with side length $2$. Let the circle with diameter AB be $\gamma$. Consider the two tangents from $C$ to $\gamma$, and let the tangency point closer to $A$ be $...
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4answers
61 views

If the median AM of a triangle ABC bisects the angle $\hat{A}$, then the triangle is an isosceles.

Can we solve the above problem using only the criteria for congruent triangles (i.e., without using the fact that the sum of the angles of a triangle is $180^\circ$)?
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1answer
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Find the Length of $QP$

Given two circles with radii $8$ and $6$ units with centers $A$ and $B$ such that $AB=12$ If $P$ is mid point of $QR$ Find Length of $QP$ My try: I assumed $A(0,0)$ and $B(12,0)$ So the equations ...
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1answer
27 views

Rectangle trapezoid

I would be very grateful if you can help me with this problem. I've constructed the median ON, N ∈ BC, and I was able to find that the triangle OCN is isosceles (height and median coincide). Probably ...
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Is there anything missing in this proof?

I came with this geometry problem and numerous lengthy solutions were proposed, so I thought there must be something missing in my solution. The problem: Given that $\angle CAB=3x$, $\angle BCA=x$, ...
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2answers
52 views

If the sides of a quadrilateral are $a,b,c,d$, prove that the area cannot exceed $(ac+bd)/2$.

MOP 1997: Let $Q$ be a quadrilateral whose side lengths are $a,b,c,d$ in that order. Show that the area of $Q$ does not exceed $(ac+bd)/2$. My solution: Without loss of generality, let $a$ be the ...
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In a cyclic $\square ABCD$, $BC, CD$ and $DA$ are three tangents of such a circle that its center is on the side $AB$. Proving that $AD + BC = AB$

In a cyclic quadrilateral $ABCD$, $BC, CD$ and $DA$ are three tangents of a circle. The center of the circle is located on the side $AB$. Prove that $$AD + BC = AB$$ Attempt: First, I thought it to ...
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27 views

Minkowski sum of a circle and ellipse

What will be the Minkowski sum of a circle and ellipse? Will it be an ellipse of the major axis (a+r) and minor axis (b+r) centered at the location of (x+y), where x is the center of the circle of ...
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1answer
36 views

Consecutive Vertices of a Quadrilateral

$D, G$ are points on the side $AB$ of $\triangle ABC$. $E$ and $F$ are points on the sides AC and BC respectively such that $DE \parallel BC,$ $EF \parallel AB$ and $FG \parallel CA$. Then $D, E, F, G$...