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.

learn more… | top users | synonyms (1)

39
votes
3answers
2k views

A generalization of IMO 1983 problem 6

Note: This question has a bounty that will expire in just a few days. Let $a,b,c$ and $d$ be the lengths of the sides of a quadrilateral. Show that $$ab^2(b-c)+bc^2(c-d)+cd^2(d-a)+da^2(a-b)\ge 0$$ ...
1
vote
1answer
1k views

Parametric Equations of an Oblique Circular Cone

I am trying to determine the parametric equations for a specific shape of an oblique circular cone with no success. Exhaustive web searchs and many texts have not been fruitful as regards ...
1
vote
0answers
234 views

References for problems related to uniformly distributed points/ arcs on circle

I am looking for references to problems related to computing the statistical properties of uniformly randomly chosen points or arcs on the unit circle, e.g.: On a unit circle, $n$ points are ...
4
votes
1answer
73 views

Existance and uniqueness of solution for a point with fixed distances to three other points

I have two sets of known points in $\mathbb{R}^2$: Four points $\mathbf{p_1}, \mathbf{p_2}, \mathbf{p_3}, \mathbf{p_x}$ , and three other points $\mathbf{q_1}, \mathbf{q_2}, \mathbf{q_3}$. I would ...
0
votes
1answer
68 views

Geometry Problem on Finding Angles

In an isosceles triangle ABC,AB=AC,P and Q are points on AC and AB respectively such that CB=BP=PQ=QA.Then prove that angle AQP=900/7
1
vote
3answers
381 views

Geometry problem on circles from a competition

Triangle $\triangle ABC$ is an equilateral triangle whose side is $16$. A circle meets the sides of the triangle at $6$ points: it intersects $AC$ at $G$ and $F$ and $|AG|=2$, $|GF|=13$, $|FC|=1$. ...
2
votes
1answer
81 views

Help on whether a geometry solution is valid.

Let $ABC$ and $AB'C'$ be similar right angled triangles with right angles at $C$ and $C'$, respectively. Let $l$ be the line between $C$ and $C'$, and let $D$ and $D'$ be the points on $l$ such that ...
4
votes
1answer
175 views

Euclidean Ramsey theory problem

Let $k\geq 1$ be given. Consider the following statement: For all (non equilateral) triangles (represented by 3 points in $\mathbb R^2$) and for all $k$-colorings of $\mathbb R^2$ there exists a ...
1
vote
2answers
59 views

Proving angle equality in a circle

Let $ABDC$ be a quadrilateral inscribed in a circle with centre $O$. If the diagonals $AD$ and $BC$ intersect at point $E$, then we have to prove that sum of angles $AOB$ and $COD$ is equal to twice ...
0
votes
2answers
204 views

Why are $\triangle ABC$ and $\triangle CDE$ similar?

Português: Porque $\triangle ABC$ e $\triangle CDE$ são similares na imagem abaixo? Sendo que $M$ e $N$ são pontos médios de seus segmentos. $AD$ e $BE$ são alturas relativas. $$$$ English: Why are ...
0
votes
1answer
231 views

A geometry problem on locus

Prove that the locus of the midpoints of the parallel chords of a circle is the diameter of the given circle.
0
votes
2answers
140 views

A plane geometry tough problem

$ABCD$ is a quadrilateral. $P,Q,R,S$ are the midpoints of $AB,BC,CD,DA$ respectively. $PR$ and $SQ$ intersect at $L$. $T$ is any point within the quadrilateral. Prove that ...
2
votes
0answers
89 views

quadrature formula for a lune

I have been reading about quadrature formulas in the complex plane. On the set $\mathbb{D} = \{|z| < 1\}$ we have $\int_{\mathbb{D}} f(z) dA = \pi f(0)$. Standard result in Harmonic functions. ...
6
votes
0answers
647 views

Minimal number of moves to construct the challenges (circle packings and regular polygons) in Ancient Greek Geometry?

In the web game Ancient Greek Geometry, there are challenges to construct regular polygons and circle packings using ruler and compass constructions. The game measures the number of line and circles ...
1
vote
1answer
174 views

Triangle inequality for an obtuse triangle

$\alpha < 45^\circ$, how to show that 1) $|AB+AC|>|DB+DC|$? 2) $|AB+AC|>|DB+DC+DA|$?
2
votes
0answers
44 views

light geometry and art

I'm a professional artist. I have a question regarding light logic. I have simplified the problem a bit. Imagine a segment in 3D space, AB of known length and a point light illuminating the ...
0
votes
2answers
349 views

Area of quadrilateral inside a triangle.

Let us consider an isoscles triangle ABC where $AB=AC=\sqrt{13}$,and length of $BC=4$.The altitude on $BC$ from $A$ meets $BC$ at $D$.Let $F$ be the midpoint of $AD$.We extend $BF$ such that it meets ...
0
votes
3answers
237 views

how to calculate the area of ​​a rhombus is in $cm^2$

How do I calculate the area of ​​a rhombus is in $cm^2$? Is the formula $\frac12 \times 17 \times 16$? Anyone can help me to solved this? I don't know the rhombus formula. Based on my exercise ...
1
vote
0answers
139 views

Bound on the angle between a vector and a subspace

Suppose you have three complex vectors $x_1$, $x_2$, and $x_3$. Define $a = \angle(x_1,x_2)$, $b = \angle(x_1,x_3)$. My question is about $c = \angle(x_1, span(x_2,x_3))$, the angle between the vector ...
0
votes
2answers
383 views

Proof of the following: How many $(n-2)$ dimensional faces from a corner of a hypercube

I asked a question earlier regarding the number of $(n-2)$ dimensional faces exiting a corner of an $n$ dimensional hypercube. (For example the number of points in a corner of a square, or the number ...
6
votes
4answers
461 views

Some theorems in euclidean geometry have incomplete proofs

I have seen that, in euclidean geometry, proofs of some theorems use one instance of the 'geometric shape'(on which the theorem is based) to proof the theorem. Like, the proof of 'A straight line ...
0
votes
1answer
242 views

Converting a distance matrix into Euclidean vector

I have a distance matrix between different elements. Now I want to calculate the Euclidean vectors that have resulted in that matrix. Is there any efficient method that can do so?
4
votes
1answer
126 views

Covering all the edges of a hypercube?

Consider an arbitrary $n$- dimensional hypercube: If we select $n - 1$ corners of that hypercube and highlight all $(n - 2)$ dimensional elements that originate from each of the corners is it ...
1
vote
1answer
1k views

Equilateral and equiangular polygon

Can we have an equilateral polygon $n \geq 5$, which is not equiangular? Ot does every odd n-gon which is equilateral must be equiangular? Is a construction of an equilateral but not equiangular n-gon ...
1
vote
1answer
53 views

form of groups of motions of tessellations

I have read from the book "Mathmatics and Its History" by John Stillwell. In Section 18.6 it is about complex interpretations of geometry. The book says: The triangle and hexagon tessellations have ...
0
votes
1answer
74 views

Can the triangle inequlity extened to show the distance inequlity of a trapezium

$AB // CD$. What are the angle conditions (acute, obtuse or right angle) of $a,b,c,d$ to be satisfied the inequality $ |AB+BC| > |CD|$? $AB,BC,CD$ are distances.
3
votes
1answer
168 views

Do we need measure theory to prove length of diameter and circumference is constant is a circle?

In high school I was taught that $\pi$ is the ratio of the length of circumference and diagonal of a circle. But is it necessary to use some measure theory machinery to define the length of ...
0
votes
1answer
231 views

parallel and normal projections

I have a vector $v$ given by $(v_x, v_y, v_z)$ which makes an angle $\theta$ with the $x$-axis. The projection of $v$ onto $x$ is given by the dot product $$v\cdot x = ...
24
votes
2answers
429 views

When is a metric space Euclidean, without referring to $\mathbb R^n$?

Normally, the Euclidean space is introduced as $\mathbb R^n$. However, I've now been thinking about how one might define the $n$-dimensional Euklidean space only from the properties of the metric. ...
1
vote
1answer
59 views

How to prove $XP = X'P$?

Two triangles $\triangle ABC$, $\triangle A'BC$ have the same base and the same height. Through the point $P$ where their sides intersect we draw a straight line parallel to the base; this line meets ...
1
vote
1answer
162 views

justifying reflection across line in beltrami-klein model

Justify the following construction of the Klein reflection A' of A across m. Let Λ be an end of m and P be the pole of m. Join Λ to A and let this line cut y (which is the circle, my note) ...
1
vote
1answer
59 views

A curious near concurrence

In a triangle $ABC$ with incircle $\omega$, excenters $I_A, I_B, I_C$, let $t_{AB}$ and $t_{AC}$ be the tangent lines to $\omega$ through $I_A$ that are closer to $B$ and $C$ respectively. Construct ...
2
votes
1answer
153 views

Trigonometry and algebra question

Given: The total length of ad + dc The lengths of each ab, bc and ...
1
vote
0answers
28 views

Trigonometry and algebra question [duplicate]

Given: The total length of ad + dc The lengths of each ab, bc and ...
5
votes
1answer
748 views

proving that a point is a center of a circle

If i want to prove that a point $O$ is a center of a circle. is it sufficient to say that if $A,B,C$ are points On the circle and $AO=BO=CO$ so point $O$ is the center because of: Through any three ...
1
vote
0answers
97 views

Lamé parameters and distance on a curved surface

I was wondering if it is possible to compute the distance between two points which lay on a curved analytical surface. The surface is defined with differential geometry formulae (position vector of ...
3
votes
1answer
81 views

can I travel in the same direction along the surface of an ellipsoid without ever returning home?

Starting from an arbitrary point on an ellipsoid, moving straight at a random direction along the surface, are you always guaranteed to come back to the starting point eventually?
1
vote
2answers
412 views

Does this proof work to prove that the greatest area of a triangle inside a circle is when the triangle is equilateral?

Does this proof work to prove that the greatest area of a triangle inside a circle is when the triangle is equilateral? I gather it doesn't because most of the proofs I've seen use derivatives etc. If ...
1
vote
1answer
5k views

Equation of a line passing through a given point, perpendicular with a vector

Find the line that goes through A(1,0,2) and is perpendicular to r = (-2,3,4) + s (1,1,2) I did a bunch of work, but I don't know if any of it is right. I erased most of it, but this is what I came ...
1
vote
2answers
75 views

Finding distance from point to line

Knowing the position of 3 points($A, B, C$) , how can I get the distance from $A$ to the line $\overline {BC}$ if I know the angle?
21
votes
5answers
967 views

Is there a deep reason why $(3, 4, 5)$ is pythagorean? [closed]

The triple $(3, 4, 5)$ is a pythagorean triple - it satisfies $a^2 + b^2 = c^2$ and, equivalently, its components are the lengths of the sides of a right triangle in the Euclidean plane. But of ...
3
votes
2answers
3k views

Deriving the formula for the volume of a sphere

A circle $x^2 +y^2 =a^2$ is rotated about the $y$-axis to form a solid sphere of radius $a$. How do you express this motion mathematically in such a way that it allows me to arrive at the formula ...
0
votes
2answers
35 views

What is the ratio of the perimeter of $OPRQ$ to the perimeter of $OPSQ$?

Area of circle $O$ is $64\pi$. What is ratio of the perimeter of $OPRQ$ to that of $OPSQ$ ($\pi = 3$)? Okay i have tried couple of things but seems its not working . Please suggest me proper ...
1
vote
1answer
217 views

equation for a line in 3D, but with shifted origin

The (unit) direction vector $r$ of a line $L$ in 3D I write in spherical coordinates, $$ r_x = \sin\theta\cos\phi \\ r_y = \sin\theta\sin\phi \\ r_z = \cos\theta $$ This line passes through the point ...
1
vote
0answers
37 views

generalizing symmetry axis of elliptical contours in 3D

The contours of the following function $f$ trace out an ellipse, $f(x, y, z) = \exp(-x^2a)\exp(-y^2b)$, where $a\neq b$ are positive, real constants greater than zero. The axis of these ellipses is ...
1
vote
0answers
87 views

Probability of a certain circular configuration

Pick each of $n$ angles , $\theta_1$ through $\theta_n$ , uniformly randomly in the range $[0,2\pi$]. Define the distance $d_{i,j}$ between $\theta_i$ and $\theta_j$ by $d_{i,j} = \min(|\theta_j - ...
5
votes
4answers
216 views

Why is it impossible to find distinct $z_1,z_2,z_3, z_4\in \mathbb C$ such that $|z_1- z_2|=|z_1-z_3|=|z_2-z_3|=|z_1-z_4|=|z_2-z_4|=|z_3-z_4|$?

A. It is possible to find distinct $z_1,z_2,z_3\in \mathbb C$ such that $|z_1-z_2|=|z_1-z_3|=|z_2-z_3|$. Answer: True B. It is possible to find distinct $z_1,z_2,z_3, z_4\in \mathbb C$ such that ...
8
votes
1answer
2k views

Sum of distances from triangle vertices to interior point is less than perimeter?

Let $M$ be a point in the interior of triangle $ABC$ in the plane. Prove $AM+BM+CM<AB+BC+CA$. The above question was posed to someone I know who is taking high-school Euclidean geometry. I'm ...
3
votes
1answer
465 views

Do the tangents of two circles define concentric circles?

Given two non-overlapping circles, $R_1$ and $R_2$. The radii of $R_1$ and $R_2$ may be different. The distance between the centers of $R_1$ and $R_2$ is defined as $x$. Draw the four tangents ...
1
vote
1answer
609 views

Finding the x-coordinate of the max point of $y = x\sqrt {\sin x} $ so that it satisfies the equation $2\tan x + x = 0$

The maximum point on the curve with equation $y = x\sqrt {\sin x} $, $0 < x < \pi $, is the point A, Show that the x-coordinate of point A satisfies the equation $2\tan x + x = 0$ I ...