geometry assuming the parallel postulate of Euclid: 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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Center and angle of complex function

Does a complex function of type $f(z)=az+b $ always have a center and angle (of rotation) or only when $b=0$ since $b\neq0$ represents a translation?
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Straight lines which join the ends of equal and parallel straight lines in the same directions are themselves equal and parallel

I have a small question regarding proposition 33 of the elements of Euclid. http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI33.html We want to prove that two lines joining equal parallels ...
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1answer
18 views

Action of the Euclidean group, generalizing linearity?

I have a vector $v \in \mathbb{R}^2$ and two elements $(A,a)$ and $(B,b)$ of the Euclidean group $E(2)$. If the relation $$[(A,a)(B,b)](v) = v$$ holds, can I say that $(A,a)(B,b)$ is the neutral ...
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On the solutions of a system of inequalities avoiding Helly's theorem

Let $a_1,b_1,\cdots,a_4,b_4\in\mathbb{R},r_1,\cdots,r_4\in(0,+\infty)$. Show that, if $\not\exists (x,y)\in\mathbb{R}^2$ such that $$ \begin{cases} (x-a_1)^2+(y-b_1)^2\le r_1\\ ...
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21 views

Be $m$ and $n$ two perpendicular lines, and …

Be $m$ and $n$ two perpendicular lines, and be distinct points $A$ and $B$ outside the lines and in the first quadrant. What is the shortest way to get from point $A$ to point $B$ by tapping the two ...
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2answers
55 views

A-Level/GCSE Geometry textbook? Geometry for STEP and MAT?

everyone. I have been looking for a book that covers the most elementary parts of Geometry, such as similar triangles, circles(arc, sector and others), Pythagorean theorem, Sine and Cosine Laws, so ...
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1answer
29 views

Side-angle-side and side-angle-angle as proved by Euclid in the Elements (Proposition 26)

I have small question regarding this proposition : http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI26.html To prove that one side is equal to another, Euclid assumes that one side is bigger ...
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153 views

What is the area of shaded region which is lies between outer and inner circle.

There is a outer circle with radius 2r and another inner circle with radius r whose center is the middle of big circle.As depicted in the following figure. Foo graph Image There is a sector of 120 ...
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14 views

Subset convex of plane

A plan of the subset is $convex$ if the segment connecting any two of its points is fully contained therein. The simplest examples of $convex$ $sets$ are the plan itself and any half-plane. Show that ...
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2answers
44 views

Hard problem about law of the cosine

I have been trying without success to prove by contradiction the following problem: Given 5 segments $x_1\leq x_2\leq x_3\leq x_4\leq x_5$ each three of which are sides of a triangle. Prove that ...
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Proofs of the three-perpendiculars theorem

I have to prove this theorem in three different ways. I have already proved it geometricaly and using vectors, but I can not think of any other way. Theorem: If PQ is perpendicular to a plane XY and ...
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1answer
12 views

Solve a convex quadrilateral with four sides and equality of two adjacent angles analytically?

Given the length of four sides of a convex quadrilateral and knowing that two adjacent angles are equal, the quadrilateral is determined. I want to know whether there's a formula representing the ...
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1answer
42 views

Performing projections with distance different to Euclidean

This is the first time I'm asking a question on math section of stackexchange, so please excuse me if this isn't the right place for such a question. I'm a programmer studying about an algorithm ...
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2answers
41 views

Denial of the 5th postulate of Euclid

I am trying to recover the denial of the Playfair's axiom but it is logical a bit strange. "To a given line and a point not on it, there is only one line through this point parallel to it". This ...
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1answer
54 views

In any triangle the angle opposite the greater side is greater.

I have a small problem with the following : http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI18.html I did understand the proof, but the proposition claims that the angle opposite the ...
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1answer
30 views

Very naive questions in elementary geometry

I was wondering whether the following questions are difficult to solve : Consider a triangle ABC (defined in euclidean geometry). Let M be inside the triangle ABC such that the triangles AMB, AMC ...
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1answer
22 views

How to show express $y $ in terms of angle $\theta$?

$ABC$ is a straight line with $AB = BC = 3$ units. $B$ is the centre of the circle with radius of $2$ units. $P$ is a point on the circle. $\widehat{B_1} = \theta$, $\widehat{A} = x$, $QC \perp AC$ ...
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28 views

What is the fraction of volume of unit hypersphere centered at one of the vertices of hypercube to that of hypercube?

consider a hyper-cube of n-dimension having a length of "r" units across each dimension. If a unit n-dimensional sphere is present at one of the vertices of the hyper-cube. what fraction of volume of ...
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43 views

Euclid's elements proposition 17. The sum of two angles in a triangle is less than 180 degres.

I have a very short question on this proposition : http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI17.html I understand the way the theorem was proved. Euclid proves that angles ABC and ACB ...
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1answer
40 views

On the definition of sphere in analytic geometry…

Last year, when I was teaching mathematics (analytic geometry) for one of my clever freands, I arrived to the definition of sphere. I said Fix $r>0$, An sphere is the set of all triples ...
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2answers
33 views

Given a triangle $ABC$, make it a point $D$ on the side $AB$.

Given a triangle $ABC$, make it a point $D$ on the side $AB$. Show that $\overline {CD}$ is smaller than the length of one of the sides $BC$ and $AC$. Ideas? The triangular inequality will not. I ...
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1answer
12 views

How do I get vectors orthogonal to the one generated by the spherical coordinate formula?

Given a formula: F : ℝ → ℝ → ℝ3 F(θ,φ) = (cos(φ)*sin(θ), sin(φ)*sin(θ), cos(θ)) what are the formulas: ...
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135 views

Hexagon packing in a circle

Suppose I want to pack hexagons in a circle, as on the drawing below (red indicates "packed" hexagons). I am wondering what is known about this problem. Specifically, I am interested in an ...
2
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1answer
55 views

An inequality about inner product in $\mathbb{R}^2$.

Let $a_i,b_i,r_i,s_i$ be positive integers for $i\in\{1,2\}$. $r_i$ and $s_i$ are non-zero for $i\in\{1,2\}$. Let $a=\left(\frac{1}{a_1},\frac{1}{a_2}\right), ...
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1answer
29 views

Geometric proof for $|| u ||^2 + || v ||^2 = \frac{1}{2}||u-v||^2 + 2||\frac{u+v}{2}||^2$

Is there an geometric proof for the following identity? $|| u ||^2 + || v ||^2 = \frac{1}{2}||u-v||^2 + 2||\frac{u+v}{2}||^2$. The norm here is normal Euclidean norm, and $u,v$ are vectors.
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Shortest path to find a highway

I remember this as a classic problem, but all Google results are video-game-related, so I guess I should ask it here: An adventurer got lost in the desert, but he knew that there was a highway ...
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1answer
50 views

Construct a regular pentagon in only 11 steps using ruler and compass. [closed]

One step is to draw a stright line or a circle (greek classical understsnding of step)
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2answers
33 views

Two lines intersect forming four angles [closed]

Two lines intersect forming four angles. If one of them is right, show that others are too straight. I am clueless how to start. Ideas?
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23 views

Diagonal of triangular bipyramid with 3 edges next to a point length 1 and orthogonal, and the lengths of three known.

I am working on a lighting system for a voxel game. It requires recursive euclidean distance calculation for successively further blocks, and the distance of each block from the light source needs to ...
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3answers
27 views

Find two points on two lines in the plane where the line between the two points go through a third point and are equidistant from that point

I have the following situation (see pic below). I have two lines $B$, $C$, in the plane, the intersection point $a$, and a point $p$. I need to find the points $b$ and $c$ along $B$ and $C$ such that ...
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30 views

Are there prerequisites to “Geometry: Euclid and Beyond” By Robin Hartshorne?

I saw that the "Geometry: Euclid and Beyond" By Robin Hartshorne seemed like a good book to learn geometry. Is this text suitable for someone who doesn't have a lot of geometry knowledge ? Thank you
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2answers
51 views

Calculating the shortest route around cylinder

If I have the following situation, what would the path look like? Where would the path go and how would I calculate it? Cylinder with diameter $10\,\hbox{cm}$ and height $5\,\hbox{cm}$. Use the ...
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1answer
39 views

Conjugate Hyperbolas.

What would be a good approach to tackle this problem. In a previous assignment I managed to show Pq=Pr. How do I show that this tangent intersects the conjugate hyperbola. Should I start by ...
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1answer
48 views

Prove that angles in a circle equal each other

Let a circle in the Euclidean plane be given, let $AB$ be a diameter, and let $CD$ be the tangent through point $A$. Let $E$ and $F$ be two points on the circle, on the same side of $AB$ as $C$, and ...
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1answer
44 views

isosceles triangles and their perpendiculars proof

Let an isosceles triangle ABC in the Euclidean plane be given, with AB being the base. DRAW the perpendiculars AD and BE from A and B onto BC and AC. Show that
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Triangle side-length problem

my problem is the following. A triangle ABC is given. P is a point on $\overline{AB}$. $k_1, k_2, k$ are the radii of the in-circles of APC, BPC, ABC. $s_1, s_2, s$ are radii of the ex-circles of ...
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58 views

Trapezoid and isosceles triangle

I have got a problem which I have to solve for my practive for an exam. Hope you can help me. An isosceles trapezoid $ABCD$ with the parallel sides $\overline{AB}$ and $\overline{CD}$ is given. ...
2
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1answer
25 views

Convex quadrangles

there is a quadrangles ABCD with $|AB| + |BC| = |AD| + |DC|$. The beam $AB$ cuts the beam $DC$ in the point $X$. The beam $AD$ cuts the beam $BC$ in the point $Y$. Now show that \begin{equation*} ...
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42 views

Finding the unit quaternion for the regular octahedron

I am given this figure of a regular octahedron and I have to find the unit quaternion. In the symmetry group of the regular octahedron what is the unit quaternion that represents a 90 degree ...
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3answers
47 views

Prove the following relation between side lengths

Let $ABC $ be any triangle , right-angled at $A$ , with $D$ any point on the side $AB$. The line $DE$ is drawn parallel to $BC$ to meet the side AC at the point E. F is the foot of the perpendicular ...
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2answers
43 views

Find the acute angles of this right triangle.

I am having trouble finding the acute angles of this triangle. O is the intersection of the medians of the triangle and $OG = \frac{1}{2}OH$. Any suggestions?
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0answers
19 views

Statue and a flag distances

Next to a flagpole is a statue that measures 9m high. The upper end of the flagpole with the bottom of the statue form an angle of 53.13 degrees to the floor, and the upper end of the flagpole to the ...
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1answer
30 views

Approximation of a Minimum Distance for two Ellipses not touch

Let $E_1$ and $E_2$ be two ellipses with centers $c_1$ and $c_2$ and semi-major axis $m_1$ and $m_2$, respectively. How could I determine a minimum $dist(c_1,c_2)$ to guarantee that two ellipses not ...
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38 views

Distance from a point to the involute of a circle

I know that the involute of circle of radius $r$ centered at $(0,0)$ is given by the following parametric form: $$\begin{cases} x(\theta) = r \big(\cos(\theta) + \theta\ \sin(\theta) \big),\\ ...
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1answer
31 views

Find all nearest points

I have two sets: $$P = \{p_1, p_2, ..., p_n\}$$ $$Q = \{q_1, q_2, ..., q_m\}$$ For each $p_i$ point I need to find all nearest points in $Q$. I.e., $$p_i \rightarrow \{ q_{i_1}, q_{i_2}, ..., ...
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2answers
43 views

what is the lowest point of a tilted elliptical plate?

I'd like to know the lowest point $z_\min$ of an ellipse with radius $r_x, r_y$ in (Euclidian) XY that's tilted in XYZ - first rotated around X axis by $\gamma$, then rotated around Y axis by ...
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1answer
59 views

In Neutral Geometry, prove that the opposite sides of a rectangle are congruent.

I'm having some trouble proving a theorem of Neutral Geometry. First, allow me to clearly state what we are allowed to assume in Neutral Geometry: Hilbert's incidence axioms Hilbert's order axioms ...
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1answer
46 views

On intervals chosen randomly within the unit circle.

Let $S = \{(x,y)\in R^2 : x^2 + y^2 = 1\}$ be the unit circle in $R^2$. Let $(X_1, Y_1), (X_2, Y_2)$ be independent, both having uniform distribution over $S$. Let $D$ denote the Euclidean distance ...
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28 views

Lines cutting regions

15 lines are drawn in a plane such that 4 of them are parallel. a. What is the maximum number of regions into which the plane is divided? b. How many of the regions are finite(bounded)? a) The ...
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1answer
103 views

If the Greeks had been four dimensional, would they have been able to derive the pi squared coefficient for the hypersphere volume without calculus?

I was reading about Archimedes' pre-calculus proof of the volume of the sphere and I realized that the trick he uses (volume of hemisphere + volume of cone = volume of cylinder) doesn't generalize to ...