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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368 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
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1answer
53 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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26 views

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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43 views

Neccesary condition for perpendicularity [on hold]

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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1answer
64 views

Find x in the triangle [on hold]

Find x, even if I turn out not
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1answer
22 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 ...
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1answer
66 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
38 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 ...
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2answers
63 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 ...
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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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20 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 ...
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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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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
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1answer
72 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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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 = ...
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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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24 views

Bisector of point and an arc [closed]

Derive an equation for the bisector of an arc C and a point P in a plane when (a)Point P not on the center and P lies inside the circle defined by the arc C (b)Point P outside the circle of arc C ...
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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 ...
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1answer
32 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 ...
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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 ...
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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 ...
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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 $\triangle ABC$ ...
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23 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 ...
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26 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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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, ...
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2answers
81 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 ...
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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 ...
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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 ...
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1answer
21 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 ...
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2answers
61 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}, ...
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1answer
43 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 ...
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17 views

Extension of Isovist concept for a point - to Isovist for a polygon

There is the concept of Isovist/Visibility polygon. They both talking about volume of space visible from a given point in space. My question: What is the algorithamic solution of this problem for a ...
4
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1answer
28 views

Determinant of matrix with unit length rows

What can be said about the determinant of a matrix when its rows (or similarly, columns) are unit vectors? Do such determinants have a geometric interpretation? For example, in the two-dimensional ...
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2answers
111 views

How to find the vertices of a particular ellipse with straightedge and compass?

In order to provide and alternative solution to a well-known problem $^{(*)}$ I would like to solve the following sub-problem in the most effective way (i.e. in the least number of steps). ...
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1answer
44 views

Calculate point of intersection line of two planes

I found some source code that I do not really understand. I will give some pseudo-code in my description to give you a better idea how the algorithm works. Basically, two planes with three vertices ...
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4answers
38 views

Linear motion in the Euclidean plane?

This is what is said in my book on linear algebra, can you please give an explanation to this? I don't quite understand the notations that are used. The set $\Bbb R^2$ can be viewed as the Euclidean ...
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46 views

Radii of the circles circumscribing the triangles $BHC,CHA,AHB,ABC$ are all equal

If $ H$ is the orthocenter of a triangle $ABC$;prove that the radii of the circles circumscribing the triangles $BHC,CHA,AHB,ABC$ are all equal.
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1answer
267 views

There is a no set which every line meets the ball

Does there exist the set of balls $X=\{B_i\subset\mathbb{R^2};i\in I\}$, satisfing following properties?(Note that the ball has a positive real radius) Let the set of all lines in plane to be $L$. ...
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189 views

An inequality about the sum of distances between points : same color $\le$ different colors?

When I was drawing some points on paper and studied the distances between them, I found that an inequality holds for many sets of points. Suppose that we have $2$ blue points $b_1,b_2$ and $2$ red ...
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1answer
27 views

Prove equivalence to the Euclidean Parallel Postulate

Show that this statement (P): The opposite sides of a parallelogram are congruent is equivalent to the H.E.P.P (Q): For every line $l$ and every point $p$ not lying on $l$ there is at most ...
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1answer
41 views

Looking for an alternative proof of the angle difference expansion

I have thought about this for a while and have no progress. Does there exist a purely Euclidean Geometric proof of the Angle Difference expansion for Sine and Cosine, for Obtuse angles?
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Question about concyclic points on the coordinate axes

If the points where the lines $3x-2y-12=0$ and $x+ky+3=0$ intersect both the coordinate axes are concyclic,then the number of possible real values of k is (A)1 ...
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External bisectors of the angles of ABC triangle form a triangle $A_1B_1C_1$ and so on

If the external bisectors of the angles of the triangle ABC form a triangle $A_1B_1C_1$,if the external bisectors of the angles of the triangle $A_1B_1C_1$ form a triangle $A_2B_2C_2$,and so on,show ...
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A movement requires 1 dimension, a rotation requires 2 dimensions, a what requires three dimensions?

Movement A zero-dimensional object cannot move. A one-dimensional object can move in one dimension (the x-axis). A two-dimensional object can move in two dimensions (the x-axis and y-axis). A three ...
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2answers
54 views

Prove line connecting intersection of tangents and opposite vertex bisects segment containing intersection of tangents and a vertex

Let $\triangle ABC$ be an isosceles triangle with $AB=BC$. Let $\Gamma$ be the circumcircle of $\triangle ABC$. Let the tangents at $A$ and $B$ intersect at $D$, and let $DC\cap\Gamma=E\neq C$. Prove ...
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1answer
98 views

One special case of Helly's theorem (for $\text{radius}=1$ circles)

There are $n$ points on the plane. Any $3$ of them can be covered with a radius $1$ circle. Prove that there is a radius $1$ circle that covers all the points. Came to this when tried to prove an easy ...
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4answers
94 views

Find the area of a triangle given the radius of its incircle and a tangential point

A friend gave me recently the following interesting problem and I would like to share a couple of solutions. Any additional contributions are welcome. A triangle $\vartriangle ABC$ is given and we ...
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A circle in the plane contains at most four lattice points?

Let $\cal C$ be a circle in ${\mathbb R}^2$ : $\cal C=\lbrace (x,y)\in{\mathbb R}^2 | (x-x_0)^2+(y-y_0)^2=r^2\rbrace$ for some constants $x_0,y_0,r$. What is the maximal number of points that can ...
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5answers
129 views

How to prove that the line perpendicular to the radius is the tangent in the calculus sense?

Let $P=(p_1,p_2)$ be a point on an semicircle and $r$ be the line perpendicular to the radius $\overline{OP}$, like the picture below. Euclid showed (Book III, Proposition 16) that $r$ does not ...