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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3answers
27 views

Percentage change using differentials.

We're given the above triangle with sides $a$ and $b$ , and area $A$. $a$ is increased by $4$% and $b$ is decreased by $3$% , we need to approximate the percentage change in the area using ...
2
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1answer
27 views

Dot Product Derviation

The dot product or inner product in Euclidean Space $A\cdot B$ has two definitions: Algebraically defined as: $$A \cdot B = \sum_{i=1}^{n}A_i \cdot B_i=A_1B_1 + A_2B_2 ... A_nB_n$$ Geometrically ...
3
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1answer
22 views

Correct Intuition? Standard Deviation and distance in $n$ dimensional space.

Basic Question Is there an intuitive explanation of standard deviation in terms of Euclidean distance in $n$ dimensional space? Longer Version of Question To begin a more detailed sketch of my ...
3
votes
0answers
25 views

Ratio between subsegments of space diagonal of a cube

Let $ABCDEFGH$ be a cube where $ABCD$ is the ground face and $E$ is about $A$, $F$ above $B$, and so on. Consider the space diagonal $AG$, and call $P$ the orthogonal projection of $B$ on $AG$. The ...
1
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0answers
23 views

Pyramid with unit sides inside a cube

Let $ABCDEFGH$ be a unit cube with base $ABCD$. Let $P$ be the top of the pyramid with base $ABCD$ and all edges of length $1$. One has a standard 2-dimensional projection of this cube on the back ...
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1answer
31 views

A geometry question on finding the area of cyclic quadrilaterals

The circumcircle of a cyclic quadrilateral $ABCD$ has radius $2$. $AC, BD$ meet at $E$ such that $AE = EC$. If $AB^2 = 2\cdot AE^2$ and $BD^2 = 12$, what is the area of the quadrilateral?
3
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1answer
18 views

In the triangle $ABC$, if $a=\frac{2(b^2-c^2)}{-b+\sqrt{b^2+4c^2}}$, prove that $3\cdot\widehat{C}=2\cdot\widehat{B}$.

Just like in the title, I have to prove that if in a triangle $ABC$ $$a=\frac{2(b^2-c^2)}{-b+\sqrt{b^2+4c^2}}$$ holds, then $3\cdot\widehat{C}=2\cdot\widehat{B}$. The denominator of the big ...
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votes
1answer
22 views

Set invariant under reflections is a ball?

Say that $A\subset \mathbb{R}^n$ is measurable and of positive, finite measure. I'm trying to see if the following is true. If $A$ is invariant under all orthogonal reflections across $(n-1)$ ...
0
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0answers
18 views

why is that a circle has the ability of maximising the area. [duplicate]

May b this question do not meet the standards of this blog but i was just calculating on things and i got stuck..actualy we were playing this game in which we were to fill students in a square and ...
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0answers
39 views

Three line segments made by intersection in harmonic progression [on hold]

I'm learning coordinate geometry in high school and have this question as a doubt. The equations of three lines are $7x + y = 16$ , $5x - y - 8 = 0$ and $x - 5y + 8 = 0$. A variable line through ...
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1answer
32 views

Euclidean geometry and $L_2(\lambda)$ space

Suppose $f,g\in L_2(I,\lambda)$ with $\lambda$ any probability measure and the norm $\| x\|=\sqrt{\langle x, x\rangle}$. Could we have the same geometric properties in this space as in the Euclidean ...
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1answer
21 views

Write $CX,AY,BZ$ in terms of $CA,CB$ and the ratios $\alpha, \beta, \gamma$?

The point $X$ divides $AB$ in the ratio $\alpha$, $Y$ divides $BC$ in the ratio $\beta$ and $Z$ divides $CA$ in the ratio $\gamma$. Write $CX,AY,BZ$ in terms of $CA,CB,\alpha, \beta, \gamma$. I ...
0
votes
1answer
20 views

What's the necessary condition for that any three vectors are parallel to the edges of a triangle in the plane?

What's the necessary condition for that any three vectors are parallel to the edges of a triangle in the plane? I answered the following: The necessary condition is that the vectors are ...
0
votes
1answer
17 views

Independence of the perimeter of a triangle and an angle formed with respect to lines tangent to a circle

Two lines passing through a point $Μ$ are tangent to a circle at the points $A$ and $B$. Through a point $С$ taken on the smaller of the arcs $AB$, a third tangent is drawn up to its intersection ...
0
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1answer
50 views

Elementary geometry question involving quadrilateral and bisector given with picture.

$ABCD$ is a quadrilateral. $m(\widehat{BAC})=48^\circ$. $m(\widehat{CAD})=66^\circ$. $m(\widehat{CBD})=m(\widehat{DBA})$. What is $\color{magenta}{m(\widehat{BDC})=x}$? Tried lots ...
3
votes
1answer
32 views

An inequality about the areas of two triangles

There is point $P$ in a triangle $ABC$. $Q,R,S$ are the symmetric of $P$ with respect to the sides $AB,BC,CA$ respectively. I have to prove that the area of $ABC$ is $\geq$ than the area of $QRS$. ...
0
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0answers
9 views

What is the perspective projection of a 3d point relative to a quarternion encoded camera?

I'm representing a camera on the cartesian space as a tuple of a 3d point (position) and a quarternion (rotation). I get the front, right and up vectors of the camera by applying the quaternion to the ...
3
votes
2answers
77 views

10 points outside a unit circle

Let $P_1$, $P_2$,... $P_{10}$ be ten points outside the unit circle centered at the origin $O$. Given that $\|P_iP_j\|\ge 1/\sqrt{2}$ for all $1\le 1<j\le 10$, find the minimum of the sum of the ...
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1answer
16 views

About the stable/invariant point sets in a plane with respect to shift/linear transformation

I'm reading Vlademir A. Zorich's Mathmatical Analysis I, meeting exercise question as following: a) A set $S \subset X$ is stable with respect to a mapping $f:X \rightarrow X$ if $f(S) \subset ...
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1answer
442 views

Which Area of mathematics can explain this?

http://i.stack.imgur.com/rij3X.png As in the image we can see that ray of light is bouncing off objects. Black ones are opaque objects and white ones are transparent objects. I want to calculate how ...
4
votes
1answer
62 views

Let the plane V be defined by $ax + by + cz + d = 0$; with $a, b, c, d \in \mathbb{R}$ and the vector $(a; b; c)$ a unit vector.

I am battling to get my mind around some of the concepts involving vectors in $3$-space. This question asks me whether the following statements are True or False: (A) The line $(a; b; c)$ is parallel ...
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1answer
87 views
+50

construct a triangle with a compass and a ruler, given $a, B, t_a$

$a,b,c$ the sides of the triangles; $A,B,C$ the angles of the triangles; $t_a, t_b, t_c$ the internal bisectors of the angles $A,B,C$. How to construct a triangle with a compass and a ruler(a ...
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0answers
44 views

Neccesary condition for perpendicularity [closed]

Triangle $\mathit{BCD}$ lies in plane $P$ and $\mathit{AD}$ and $\mathit{DC}$ are perpendicular ($A$ is the top of the prism $\mathit{ABCD}$) $\mathit{AD}$ is perpendicular to $P$ if which of the ...
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votes
1answer
66 views

Find x in the triangle [closed]

Find x, even if I turn out not
1
vote
1answer
25 views

Expression of reflection isometry in the complex plane

Using the fact that an anti-displacement in the plan has the form $$f(z) = a \overline{z} + b$$ I have done some computation to find the reflection about the line passing through two points $P$ and ...
0
votes
1answer
67 views

Manifold that is not a Euclidean space

I just started reading a textbook, and it keeps saying that an $n$-dimensional manifold is a topological space with the same local properties as Euclidean $n$-space. I don't really understand what is ...
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2answers
39 views

$AB=AC$, $BD$ is the angle bisector of $\angle B$ , find $\angle A$

Let $ABC$ be an triangle, $AB=AC$. $BD$ is the angle bisector of $\angle B$, $BD$ intersect $AC$ at point $D$, and $AD=BC+BD$. show that: $\angle BAC=20^\circ$ Well, If $\angle BAC=20^\circ$, I ...
5
votes
2answers
66 views

Trapezoids in a square

Good day As part of a problem I need to show that AB is parallel to CD, with the given info on the image. All the segments marked red are equal, all 1-stripe grey equal etc. I'd like to prove ...
0
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1answer
39 views

A rectangle with a triangle

ABCD is a rectangle. From 'C' two lines are drawn to meet AB and AD at E and F respectively (here AB and AD are not produced). From B, a line is drawn to meet CE, DE and AD at G, H, and F ...
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0answers
23 views

Subgroup of the orthogonal group generated by reflections

What is the subgroup of the orthogonal group $O(n)$ generated by relections around $k$-dimensional hyperplanes? For $k=n-1$ it is $O(n)$ by the Cartan's theorem. For $k=0$ it is $\{\pm Id \}$. What ...
0
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2answers
45 views

A question on tangent circles and finding the angle between the lines

Two circles of radii in the ratio $1:2$ touch each other externally. The center of the small circle is '$c$' and that of the bigger circle is '$D$'.The point of contact is $A$. $\overline{PAQ}$ is a ...
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votes
0answers
31 views

Triangle inequality $ax ≥ br + cq$

I got stuck on this problem : Given a triangle (△ABC) of sides $a$, $b$ and $c$, let $O$ be a point inside △ABC. Let $D$, $E$ and $F$ be points on sides a, b and respectively c such that $OE ⊥ ...
4
votes
1answer
74 views

2011 USAMO Problem 3, Hexagons.

In hexagon $ABCDEF$, which is nonconvex but not self-intersecting, no pair of opposite sides are parallel. The internal angles satisfy $\angle A = 3\angle D$, $\angle C = 3 \angle F$, and $\angle E ...
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votes
1answer
43 views

Geometry - Trapezium and its properties

In the trapezium $ABCD$ , $AB$ is parallel to $CD$ and $O$ is the intersection of $AD$ and $BC$. The line $PS$ is drawn through $O$ in such a way that $PS$ is parallel to $DC$. If $AB = 20$ AND $CD = ...
2
votes
3answers
40 views

Triangle area inequalities

I've got stuck on this problem : Proof that for every triangle of sides $a$, $b$ and $c$ and area $S$, the following inequalities are true : $4S \le a^2 + b^2$ $4S \le b^2 + c^2$ ...
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0answers
23 views

Non-orthogonal space

What does the angle between two non-orthogonal basis denote? Is it correlation or some measure of dependence. Does that mean that coordinates of a point if moved in the direction of one axis also ...
1
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1answer
33 views

$S \leq \frac{(a+b)(c+d)}4 $

I got stuck on this problem: Given a convex quadrilateral of area $S$ and sides $a$, $b$, $c$ and $d$, prove that: $$S \leq \frac{(a+b)(c+d)}4$$ What I've done so far was to proof that ...
0
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0answers
20 views

Proof of Menelaus using areas

I've tried to proof Menelaus' theorem using areas, but I've didn't figure out how. Some suggestions would be appreciated. Menelaus' Theorem states : Given a triangle ABC and a transversal ...
3
votes
3answers
81 views

Extension of metric definition to two sets

The standard definition of a metric, is a function $d: X\times X \to \mathbb{R}$. What is a sensible/common extension of a metric/pseudometric to $\tilde d: X\times Y \to \mathbb{R}$, i.e. distance ...
2
votes
2answers
43 views

Quadrilateral's area problem

I have some troubles with this problem : Let $ABCD$ be a convex quadrilateral. $M$, $N$, $P$ and $Q$ are the midpoints of the sides $AB$, $BC$, $CD$ and $AD$. $AN$, $BP$, $MD$ and $CQ$ are ...
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1answer
79 views

Geometry - angle bisector, circumcircle: SL olympiad

I tried this problem as much as I can, but I got nothing. This is a Sri Lankan mathematical olympiad problem. Let $P$,$Q$ be points on the sides $AB$ and $AC$, respectively, of a $\triangle ABC$ ...
2
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0answers
26 views

Collinearity problem (Newton-Gauss line)

I had some troubles with this problem : Let $ABCD$ be a convex quadrilateral. $M$ and $N$ are the midpoints of the diagonals $AC$ and $BD$. The sides $AB$ and $CD$ are extended until they ...
0
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0answers
27 views

Easy question, hard solution: find the area about a domain in the plane?

We want to find the area of a domain with piecewisely smooth boundary by using the coordinates $(p,\theta)$ of the random line: It has been known that every straight line $\ell$ on $R^2$ can be ...
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votes
2answers
31 views

Given two rectangles A and B and their dimensions, is a there test for lengthA<lengthB, widthA<widthB [closed]

If we have two rectangles and their dimensions, is there a mathematical test to simultaneously compare the two numbers. For examples, if we wanted to compare the area, we would compare the products, ...
9
votes
2answers
83 views

Finding tangents to a circle with a straightedge

There is a geometric construction that I heard years ago and I still haven't figured out why it works despite several attempts. Playing with pen, paper and GeoGebra makes me confident that it does ...
0
votes
0answers
13 views

Surface Area of a Self-Intersecting Spindle Torus in Flat Euclidean space

Does the usual formula for a torus from Pappus's theorem apply to self-intersecting spindle tori? To avoid any ambiguity in defining a notion of a surface area for the self-intersecting torus - I ...
0
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1answer
25 views

3d parametric spiral to 3D goldean mean spiral

I know I can create a 3d parametric spiral with the formula below but How can I do the same thing with goldean spiral? I looked at https://en.wikipedia.org/wiki/Golden_spiral but I don't see how to ...
0
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1answer
24 views

Vector norm - Understanding the definition of the unit sphere

If $\|x\|=1$ just means the vector $x$ has length one - Then why is the unit sphere defined as $S=\{x\in X| \quad \|x\|=1\}$? let $X$ be a normed linear space with the Euclidean norm, then letting ...
2
votes
2answers
63 views

Triangle Area problem

I've been trying to solve the following: Let $ABC$ be a triangle with sides $a, b $ and $ c$, inradius $r$ and exradii $r_a, r_b$ and $r_c$. If $A'B'C'$ is another triangle with sides $\sqrt{a}, ...
4
votes
1answer
45 views

Generalising Plane Isometries to $\mathbb{R}^3$

Firstly, I DO NOT WANT PROOFS OF ANY OF THESE THEOREMS, as I wish to prove them myself. However, I would like to know the correct generalizations to $\mathbb{R}^3$ of the following theorems: An ...