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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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Which double cone splits a sphere into two equal volumes?

Consider a double cone (like a past and future light cone) centered at the origin. Now imagine a sphere centered at the origin. What angle of the slope of the double cone makes it so that the it ...
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3answers
83 views

Intuitively, why should I expect a circle in the complex plane from the equation $\left|\frac{z-1}{z+1}\right| = c$?

I know how to prove that : ($c \in [0,1[$) $$C = \{z \in \mathbb{C}: \left|\frac{z-1}{z+1}\right| = c \}$$ is circle in the complex plane. To do so we can for example write $z = x+iy$ and use the ...
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39 views

Is an apeirogon contained in the euclidean plane?

The question is self-explanatory. I think it is not, because when I create an apeirogon I am not using things that euclidean geometry allows. But then what exactly are the Hilbert Axioms that are ...
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62 views

In square $ABCD$ with side $1$, points $E$, $P$, $F$ are the midpoints of $AD$, $CE$, $BP$. What is the area of $\triangle BFD$?

I'm not sure if my solution to this Olympiad Geometry question is valid. Let $\square ABCD$ be a square with side length $1$. Let $E$, $P$, $F$ be the midpoints of $AD$, $CE$, $BP$, respectively. ...
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1answer
36 views

Inscribed circles radius

Let ABCD parallelogram. The inscribed circle in triangle ABD is tangent to BD in E. Show that $$\frac{r_{DEC}}{r_{BEC}}=\tan (\frac{1}{2}\angle ACD)\cdot \tan (\frac{1}{2} \angle ADB)$$ What I have ...
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2answers
50 views

What is sufficient to prove Kiselev's Geometry #82?

I am having difficulty realizing what would be sufficient to prove problem #82 asked in Kiselev's Geometry Book I. 82.* On one side of an angle $A$, the segments $AB$ and $AC$ are marked, and on ...
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50 views

Quaternion product of three vectors: meaning of vector part?

$\newcommand{\i}{\mathbf{i}} \newcommand{\j}{\mathbf{j}} \newcommand{\k}{\mathbf{k}} \newcommand{\a}{\mathbf{a}} \newcommand{\b}{\mathbf{b}} \newcommand{\c}{\mathbf{c}} \newcommand{\R}{\mathbb{R}}$If ...
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1answer
38 views

How to solve this vector problem involving more than one unknown?

This is an exercise I came across while tutoring high school physics. I am posting this as an "answer my own question." Kyle suspends a 12340 N moose from two trees as shown below. What is the ...
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3answers
41 views

Intriguing geometry problem regarding isogonal lines

A line $r$ contains the points $A,B,C,D$ in this order. Let $P\notin r$ such that $$\angle APB=\angle CPD$$ Denote furthermore by $G$ the intersection of the angle bisector of $\angle APD$ and $r$. ...
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In acute $\triangle ABC$, show $DE+DF \leq BC$, where $D$, $E$, $F$ are the feet of the altitudes from $A$, $B$, $C$, respectively.

Let $\triangle ABC$ be an acute angled triangle. The feet of the altitudes from $A$, $B$, and $C$ are $D$, $E$, and $F$, respectively. Prove that $$DE+DF \leq BC$$ and determine the triangles ...
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76 views

Side of the equilateral triangle

I tried very much but since tomorrow is my exam, i cannot risk it. The following is a geometry problem, which i have tried very much but could not grasp a solution. I think that i require pythagoras'...
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1answer
30 views

How to layer objects - geometry?

I'm developing a kind of perspective based 2d/3d game. I've got an X- and an Y-axis like I've displayed in the image below. To my question: I've got a bunch of objects (marked with "1" and "2") on ...
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2answers
39 views

Three equilateral triangles form a hexagon [on hold]

As I posted yesterday, I was learning about vectors yesterday. I know how to add and subtract them, but I can’t multiply yet. So here is an extra problem from my teacher I need help with: Given a ...
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+50

Find a point $X$, in the plane of regular pentagon $ABCDE$, that minimizes $\frac{XA+XB}{XC+XD+XE}$.

Find such a point $X$, in the plane of the regular pentagon $ABCDE$, that the value of expression $$\frac{XA+XB}{XC+XD+XE}$$ is the lowest. I tried using Ptolemy's theorem but don't know how to make ...
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how to obtain peripheral recangle of arbitrary ellipse?

Suppose have arbitrary ellipse with center $(x,y)$ and its radius $(a,b)$. I want obtain rectangle that sides tangent of peripheral ellipse. the below image describe issue :
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1answer
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Proving Hilbert's Axioms as Theorems in $ℝ^n$

In KG Binmore's "Topological Ideas" he says The geometric terms which appear in Hilbert's axioms are the words point, line, lie on, between and congruent. To show $\mathbb{R}^2$ is a model for ...
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4answers
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Distance between the origin (0,0) and a line y = ax +b [on hold]

Derive a formula for the Euclidean distance between the origin $(0,0)$ and a line $y = ax + b$, where $a$ and $b$ are arbitrary constants.
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1answer
29 views

Coordinate-free proof that two points are diametrically opposed

Let $c$ be the center of a circle with radius $r > 0$ and let $a$ and $b$ be two points at the circle. If there exists $t \in [0,1]$ such that $c = (1-t)a + tb$, then $d(a,b) = 2r$. I'm working ...
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0answers
35 views

Is classical Euclidean geometry Turing complete?

By "classical Euclidean geometry" I don't mean the study of $\mathbb R^2$ with the Euclidean metric, or a model of Hilbert's Grundlagen axioms, or the study of those properties of $\mathbb R^2$ that ...
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3answers
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Does it exist an approximation using sums and a way to do the computation with more accuracy both by hand to find the length of a spiral?

I have found this riddle in my book and so far which may require the use of calculus (integrals) to which I'm familiar but not very savvy with it. Since, I've not yet come with an answer. I wonder if ...
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1answer
22 views

Prove that: BD/DC = AR/AS

In a triangle ABC, internal bisector of angle A intersects side BC at D. R and S are circumcentres of triangle ABD and triangle ADC respectively. Then prove that, BD/DC = AR/AS.
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Finding angles of $\triangle ABC$: $CA=CB$, $BD$ is angle bisector and $BD+DC=AB$.

Triangle $ABC$ is isosceles ($CA=CB$). $BD$ is angle bisector of $\angle B$. Find angles of triangle $ABC$ if $BD+DC=AB$. Actually, I have a solution but I don't like it too much: If you apply law ...
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Geometry and the vertices of the Birkhoff polytope

The Birkhoff polytope $P(n)$ is defined to be the points in $\mathbb{R}^{n^2}$ which correspond naturally to $n \times n$ doubly stochastic matrices. Is it possible to prove that the vertices of the ...
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2answers
35 views

Proving two lines are parallel with intersections and midpoints

To prove : Fix points $A,C$ and set point $B$ to be the midpoint of segment $AC$. Fix point $Y$ (anywhere) and consider an arbitrary point $X$ on line $YB$. If $P$ and $Q$ are the intersection ...
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4answers
37 views

If AH and BG are angle bisectors, how would I find IJ?

Diagram I've tried finding it, but it just doesn't seem to come out. I found $$GC=\dfrac{4}{3}$$ $$AG=\dfrac{5}{3}$$ and $$GB=\dfrac{4\sqrt{10}}{3}$$ I really don't know what to do from here, could ...
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2answers
52 views

if triangle ABC has incenter I how is it possible to show the circumcenter of triangle BIC lies on the circumcircle of triangle ABC?

if triangle ABC has incenter I how is it possible to show the circumcenter of triangle BIC lies on the circumcircle of triangle ABC? D is the circumcenter of BIC
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1answer
32 views

Get direction vector of a line from a point and its rotation.

The title pretty much says what I need. Some details about my problem: I have a cylinder. I have a point outside this cylinder. I want to find the direction from said point to the line that passes ...
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2answers
27 views

Find a ratio of triangle's height segment

Given a right angle triangle ABC (C = 90) and a median AM. CD is the height of the triangle ...
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0answers
31 views

Ratio of Volume of sphere to Volume of cube

I was told in my class that the ratio of the area of a circle to area of a square should be greater than the ratio of the volume of a sphere to volume of a cube. But, I am not able to show this. For ...
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2answers
205 views

What's the precise definition of coordinates in Euclidean space?

I have a loose understanding of what coordinates are, but not something rigorous or concrete. For example take the statement of this result below When the author says let $x = (x^1, \dots x^n)$ ...
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1answer
27 views

Euclidean space: $k$ points in $\mathbb{R}^n$

Consider $k$ points $p_1,\dots,p_k$ in $\mathbb{R}^n$. Then, $\forall i,j=1,\dots,k: p_i + \operatorname{span}(p_1-p_i,\dots,p_k-p_i) = p_j + \operatorname{span}(p_1-p_j,\dots,p_k-p_j).$ Assume that $...
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2answers
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Diagonal calculation in a 3D square

I have the information as shown in the following image: i.e.: all sides (AB, BC CD and DA) and one diagonal (BD) and the height difference between the point A and the points B/C/D (855.35mm). How ...
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1answer
55 views

Show that if $AR$ intersects midperpendicular of $MN$ at $X$, then $X\in(I)$

Triangle ABC has an incircle $(I)$ which contacts $BC,CA,AB$ at $D,E,F$. Let $BP,CQ$ be bisectors of $\angle ABC,\angle ACB$ ($P \in AC,Q\in AB$). Line $AI$ intersects circle $(I)$ at $J$ (point $J$ ...
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0answers
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Scalar product of two points coordinates

In one of the algorithms I encountered an formula to calculate a scalar number from 2 points. Given Point 1 has coordinates $(x1,y1)$ and Point 2 has coordinates $(x2, y2)$ the formula was following $...
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3answers
83 views

Show that $MNPQ$ is a square

Let $ ABCD $ a quadrilateral s.t. $AC=BD $ and $m (\angle AOD)=30°$ where $O=AC\cap BD $. Let $\triangle ABM, \triangle DCN, \triangle ADN, \triangle CBQ $ equilateral triangles with $Int (\triangle ...
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1answer
14 views

Isometries of the Plane, Euclidean space $R^3$ and isometries of the Platonic polyhedra.

I want to study the isometries of plane, Euclidean space, and the platonic polyhedra. I am new to this topics. Can any one suggest books that contain these topic with details and basic explanation ...
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1answer
71 views

What is the value of AD+CD where ABC is an isosceles triangle, D bisects angle ACB, BC = 2017 unit?

$ABC$ is an isosceles ($AB = AC$) and $\angle A = 100^{\circ}$. An point $D$ is on $AB$ so that $CD$ angle bisector $\angle ACB$. If $BC= 2017$ then calculate $AD+CD$. Source: Bangladesh Math ...
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1answer
39 views

Incircle of a triangle

In the above image, it says $$AE = \frac{bc}{c+a}$$ and $$AF = \frac{bc}{a+b}$$ But $AE$ and $AF$ are tangents from $A$ to the incircle. As tangents on a circle from a given point are equal, $AE=AF$ ...
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2answers
41 views

Why does having alternate interior angles congruent, etc., prove that two lines are parallel?

I know that if two lines are parallel and there is a transversal crossing both, the alternate interior angles are congruent, alternate exterior angles congruent, etc. etc. According to the geometry ...
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0answers
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Inscribing tangential circle in non-tangential circles

Given the positions ($p_1$, $p_2$, $p_3$) and radii ($r_1$, $r_2$, $r_3$) of three circles that are not pairwise tangential, how do you calculate the location and radius of the circle that is pairwise ...
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Find area of quadrilateral in triangle. [closed]

What is the area of $HIJK$ quadrilateral, if the area of $ABC$ triangle is $70$, $BE=ED=DA$, and $BF=FG= GC$?
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2answers
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Construct a quadrilateral, not a parallelogram, in which pair of opposite angles and a pair of opposite sides are equal. [duplicate]

I want to construct a quadrilateral which is not a parallelogram, in which a pair of opposite angles and a pair of opposite sides are equal. I tried drawing one, but I am not able to. Please help. (...
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0answers
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Solving doubleintegral $\int\int f(g(x,y)) dy dx$ over annulus $D$ in $R^2$ where $g$ is Isometry

I want to solve certain triple integral $\int\int f(g(x,y)) dy dx$ over annulus $D$ (concentric about origin) in $R^2$ where $g$ is an Isometry of a plane and $f$ is a given function. Because of the ...
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2answers
40 views

The size of the biggest square that can be inscribed within a circle

(a) Is the size the biggest square that can be fit inside a circle always $2r^2$? (b) How do you do when you show that is indeed the case? I want to compare my boyfriend's proofs with that of ...
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1answer
28 views

Is punctured disc a Lipschitz domain?

I don't quite understand how to apply the definition (Understanding Lipschitz domain) of Lipschitz domain. My question is about annulus (of which punctured disc is a special case). Is annulus a ...
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1answer
40 views

Maximizing the area of a cyclic trapezoid whose long base is the circumdiameter. Non-trigonometric solution?

A half circle with a radius of R encompasses an isosceles trapezoid such that the large base of the trapezoid is the diameter of the circle encompassing it. In terms of R, what is the length of ...
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Effect of plane isometry on punctured disc

I want to confirm if I am correct. Due to the rotational symmetry of open unit disc $D$, it suffices to study only translations on $D$ out of the four plane isometries? This is because, every other ...
2
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1answer
60 views

Geometrical problem in Newton's “Principia”.

Let VQPA be the circumference of the circle, S the given point toward which the force tends as to its center, P the body revolving in the circumference, Q the place to which it will move next, and PRZ ...
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a problem about the incircles of two triangles that the orthocenter formed.

See below diagram. $H$ is the orthocenter of an acute triangle $ABC$ where $AB \neq AC$. The circle centered at $I$ and the circle centered at $J$ are the incircles of triangles $ABH$ and $ACH$. $XY$ ...
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Equal sums of surfaces

The inner points $X$ and $Y$ in the convex quadrilateral $ABCD$ are such that $\angle ABX=\angle CBY$, $\angle BCY=\angle DCX$, $\angle CDY=\angle ADX$ and $\angle DAY=\angle BAX$. Prove that $$S^{}_{...