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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2
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1answer
516 views

Internal polygon formed by drawing diagonals in a regular polygon

In an n-sided (n>4) regular polygon, label the vertices {0, 1, ..., n-1}. For each vertex i, draw a pair of diagonals: from i to (i+2) mod n and from i to (i-2) ...
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0answers
15 views

Find a point along the line created by two points based on distance [on hold]

Let's say I have two people on two points on a 2 dimensional plane. Person B is 400 units away from Person A. But Person A wants to be always 950 units away from Person B, preferably moving back in a ...
8
votes
3answers
72 views

Does a set of $n+1$ points that affinely span $\mathbb{R}^n$ lie on a unique $(n-1)$-sphere?

In $\mathbb{R}^2$ every three points that are not colinear lie on a unique circle. Does this generalize to higher dimensions in the following way: If $n+1$ element subset $S$ of $\mathbb{R}^n$ does ...
3
votes
1answer
45 views

A straightedge and compass construction: $\left(G,I,Q_a\right)$

Construct $ABC$ with straightedge and compass, given $G,I,Q_a$. $G - $ the intersection point of medians; $I -$ the center inscribed circle; $Q_a -$ point of tangent inscribed circle to the side ...
10
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2answers
189 views

about a ninth-grade geometry problem

My brother asked me this problem, and he is studying ninth-grade. I can't solve it using primitive tools of pure geometry. Hope someone can give me a hint to solve it. Thanks. Given a circle $(O, ...
0
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1answer
36 views

Problem on circles, tangents and triangles

Let $c_1,c_2,c_3$ be three circles of unit radius touching each other externally. The common tangent to each pair of circles are drawn (and extended so that they intersect) and let the triangle formed ...
1
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1answer
26 views

Efficient assignment of tetrahedron's chirality

Suppose we have a regular tetrahedron delimited by four points $A_{1}, A_{2}, A_{3}, A_{4}$. There are 24 permutations of vertices, but there are only two distinct terahedra that cannot be ...
0
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2answers
51 views

How to find the tangency condition for this circle geometry problem?

Suppose I have a circle $C$ of radius $1$, and I have a chord of this circle, of given length $l$. The chord makes a known angle $\theta$ with the tangent to the circle. I position a smaller circle $...
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0answers
37 views
+50

supporting function and halfspace (definition)

we've defined the following: supporting function: Let $P$ be a convex polygon in $E^d$ (euclidean vector space). Then the supporting function is defined as $h_P: S^{d-1} \to \mathbb{R}$ by $h_P(u) :=...
0
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1answer
80 views

Collinearity of triangle vertices on circular paths

Suppose as a function of time, the vertices of a triangle move with constant speed along circular trajectories: $$ \vec p_i(t) = a_i \left( \begin{array}{c} \cos(b_i t+ c_i)\\ \sin(b_i t+ c_i) \end{...
2
votes
1answer
59 views

Brocard Angles proof by Sine and cosine formulae.

The angles denoted by $\omega$ are the Brocard angles. Recently i came to know about the Brocard Angles and also their property i.e $\cot{\omega}=\cot{A}+\cot{B}+\cot{C}$. In my previous question I ...
0
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1answer
32 views

Exercise on three planes meeting in a line.

In $R^3$, Given the plane $\pi : ax + by +cz + d = 0$ and the planes $\alpha : y + z = 2, \quad \beta: x - y + z = 0$ . Do there exist values of $a,b,c,d$ s.t. the three planes meet two by two in a ...
8
votes
1answer
1k views

How to plot $N$ points on the surface of a $D$-dimensional sphere roughly equidistant apart?

Let's say I have a $D$-dimensional sphere with a radius $R$. I want to plot $N$ number of points evenly distributed (equidistant apart from each other) on the surface of the sphere. It doesn't matter ...
0
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0answers
25 views

How does inversion affect the angle subtended by a circular arc?

Say that I describe a circular arc $A\subseteq\mathbb{R}^2$ using an ordered triple $(p_1,p_2,c)$, where $p_1,p_2$ are the endpoints of the arc and $c$ is its center. (Technically this also describes ...
0
votes
1answer
681 views

Find locus of points relating to an ellipse

I would like to find the equation of the following locus. For a big circle C centered at (0,0), the locus of points that the sum of distances to Y-axis and to C is 1, say in the first quadrant, is ...
1
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0answers
66 views

Can one define 'geodesic' solely in terms of the betweenness relations among the points on that geodesic?

In the Euclidean plane (though I assume the following result can be generalized to any Euclidean n-space), Tarski showed that one can define what it is to be a straight line solely in terms of the ...
2
votes
3answers
172 views

Prove triangle similiarity by given expression

I am working on the following problem, but I can't seem to figure it out. The length of the sides in the triangle $T_1$ are $a_1$, $b_1$ and $c_1$. The length of the sides in the triangle $T_2$ ...
6
votes
2answers
164 views

Triangle in perspective to a given triangle but similar to another

Is it always possible to construct a triangle that is in perspective to a given triangle and have it also be similar to a different given triangle? If you create a triangle in perspective to another, ...
1
vote
1answer
24 views

Collinear Points in 3-Dimensions

The points A(3, -1, z), B(1, 2, 6), and C(x, 8, 14) are collinear. Find the values of x and z. I have tried finding common ratios between the points, but no common ratio is possible, I have a feeling ...
3
votes
2answers
63 views

Homeomorphism from $S^1\backslash(0,1)$ to $\mathbb{R}$

I am trying to derive a bijection between $S^1\backslash{(0,1)}$ and the real line, but I am stuck on using the most obvious way Let the top point of the circle be $(0,1)$, and the blue line hits ...
1
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2answers
23 views

Dimension of the span of two parallel lines in $R^4$.

I am asked if the following question is true or false: Let $r,s$ be two parallel lines in $R^4$ then the dimension of $Span(r \cup s)$ is strictly less than $3$. I think this is true because two ...
4
votes
4answers
366 views

Find the sum of angles without trigonometry?

I have found the sum it's $180$ but using right triangle and sine theorem.
1
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1answer
22 views

3D Geometry concurrency problem

$ABCD$ is a tetrahedron. Let $K$ be the center of the incircle of $CBD$. Let $M$ be the center of the incircle of $ABD$. Let $L$ be the centroid of $DAC$. Let $N$ be the centroid of $BAC$. Suppose $...
0
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1answer
55 views

Hyperbolic plane shrinking

A very small area of the hyperbolic plane looks more Euclidean as the curvature approachs 0. Any more evidence? Or reference would help? Thanks
7
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1answer
111 views

The grey area is equal to the white area

Problem. Show that the sum of the areas of the white regions is equal to the sum of the areas of the grey regions. All the angles between consecutive chords are $45^\circ$. A solution (not totally ...
0
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0answers
21 views

Piecewise linear curve where the closest vertex always belongs to closest edge

Take a piecewise linear curve $L$ in Euclidian space, i.e. a an ordered set of points $P$ sequentially connected by straight lines $l_{i}$, each defined by two points $p_i$ and $ p_{i+1}$. Some such ...
0
votes
2answers
28 views

Proof - Elementar Geometry (parallelogram)

Prove that by connecting midpoints of adjacent sides of a quadrilateral we get a parallelogram. I'm having problems with this piece of work for some time so decided to ask for help here. Though I'm ...
2
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0answers
23 views

I have a convex hull (generated from a library) in 3D. I only have the vertices. How do I compute the volume of the hull.

I have a library (quickhull in C++) that I am using to create a hull from a set of points. I am able to see the vertices of the hull but not the facets. I would like to compute the volume of the hull. ...
0
votes
1answer
1k views

calculating volume of a horizontal cylindrical tank from depth

Has anyone found a good formula to convert a depth of fluid to a volume remaining in a tank for a cylindrical tank laying horizontal? (with or without half sphere end caps) Much Thanks!
-1
votes
2answers
126 views

Golden Ratio Symphony in a 5x5 Grid & Circle: Is this golden ratio construction derivative of other constructs? Is it novel? [closed]

Below please enjoy a Golden Ratio Symphony in a 5x5 Grid & Circle: Is this golden ratio construction derivative of other constructs? Is it novel? The below construction is created by beginning ...
0
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0answers
32 views

Intersection over union optimization

Let $\mathcal R \simeq \mathbb R ^4$ be a set of all rectangles parametrized by $(x, y, w, h)$ -- coordinates of center and length of edges. How can I solve the following optimization problem: \begin{...
16
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2answers
429 views

About the ratio of the areas of a convex pentagon and the inner pentagon made by the five diagonals

I've thought about the following question for a month, but I'm facing difficulty. Question : Letting $S{^\prime}$ be the area of the inner pentagon made by the five diagonals of a convex pentagon ...
3
votes
0answers
23 views

Cutting a pie into n equal area pieces with the minimum distance of cuts. [duplicate]

Suppose we are to cut a unit circle into n equal area pieces. We can cut curves if we wish. What is the minimum distance we must cut? What strategy minimises this distance? Note: The shape of the ...
0
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0answers
47 views

Draw a line between an observer and the current direction of the sun

My goal is to draw a line between an observer and the current direction of the sun. Given the observers coordinates (Lat, Lon) of (51.50442, -0.08630) a North of (90, 0), an Azimuth of 270 degrees ...
0
votes
1answer
39 views

Prove concurrency in a triangle

If a circumference cuts a triangle $ABC$ at its sides $BC$, $CA$ and $AB$ at points $P, P'; Q, Q'; R, R'$; respectively (so twice on each side, and if $AP, BQ$ and $CR$ are concurrent (intersect at a ...
14
votes
4answers
530 views

If any triangle has area at most 1 , points can be covered by a rectangle of area 2.

I am working on this problem for some time, and I am not able to finish the argument: There is a finite number of points in the plane, such that every triangle has area at most 1. Prove that the ...
0
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1answer
18 views

Analytical Geometry medial triangle

The median $AB_1$ meets the side $A_1C_1$ of the medial triangle $A_1B_1C_1$ and $CP$ meets $AB$ in $Q$ show that $AB=3AQ$. I tried to use Ceva's theorem but couldn't do that as according to Ceva's ...
0
votes
2answers
107 views

What is the size of the angle $\angle AMC$? [duplicate]

Suppose we have a triangle $\triangle ABC$ where the size of two angles are given: $\angle B=15^\circ$ and $\angle C=30^\circ$. We draw the median $AM$, so now what is the size of angle $\angle AMC$? ...
0
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0answers
35 views

Relating fibonnaci sequence, lucas numbers and golden ratio to make a research question?

I am planning to write a high school level maths essay of approximately 4000 words. I do find Fibonacci sequence, Lucas numbers and Golden ratio amazing and want to research further on them, the thing ...
7
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1answer
53 views

Reference Request: Complete Proof of Braikenridge–Maclaurin Theorem

Where can I find a reference to a complete proof of the Braikenridge–Maclaurin theorem, which is stated as: If the three pairs of opposite sides of (an irregular) hexagon meet at three collinear ...
1
vote
1answer
16 views

Existence of non obtuse angle of n+2 vectors in n-dimensional euclidean space.

There are n+2 distinct vectors $v_1,v_2,v_3,\cdots ,v_{n+2}$ in n-dimensional euclidean space. Prove that there must be a integer pair of $(i,j)$ which satisfies $1\leq i<j\leq n+2$, and $dot(v_i,...
1
vote
1answer
78 views

Geometry-Triangle

Let $ABC$ be a triangle with $DAE$, a straight line parallel to BC such that $DA=AE$. If $CD$ meets $AB$ at X and $BE$ meets $AC$ at $Y$, prove that $XY$ is parallel to $BC$ I tried to use the angle ...
1
vote
1answer
21 views

Geometry, triangle incenter problem

I is the incenter of triangle $ABC$. $X$ and $Y$ are the feet of the perpendiculars from $A$ to $BI$ and $CI$. Prove that $XY$ is parallel to $BC$ I tried to use the angles $AXI$ and $AYI$ to prove ...
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vote
1answer
39 views

Derive a relation between angles A,B and C

Derive a relation between angles A,B and C (do not use other angles in the final relation): I have tried to use two theorems in triangles(external angle and complement angles),but no success! It ...
8
votes
2answers
170 views

A challenging straightedge and compass construction

Three points $A,O,B$ are given, and $0<\theta=\widehat{AOB}<\frac{\pi}{3}$. It is known that there are two points $A',B'$ on the segments $OA,OB$ such that $$ BB'=B'A'=A'A $$ holds. How ...
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0answers
20 views

Does the collection of algebraic/number-theoretic methods applied to Euclidean Geometry have a name?

I am currently writing an essay on the history of geometry. To educate myself on the subject, I sometimes read the following Wikipedia article on the history of Euclidean Geometry. It seems to me that,...
3
votes
1answer
27 views

Is a shape 'polarizable'?

Given a point $p$ inside a shape $S$ described as an $n$-vertex polygon, let us say that $S$ is polar with respect to $p$ if S can be described by a polar equation $r(\theta)$ with $p$ as the origin. ...
1
vote
1answer
34 views

Can circumscribing a circle around a polygon prove that the sum of the interior angles of an n-sided polygon is $180(n - 2)$?

I am trying to create my own proof that the sum of the interior angles in a regular polygon is $180(n - 2)$, where $n$ is the number of sides in the polygon. I have seen these proofs for this formula, ...
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vote
2answers
761 views

Do collinear lines or overlapping collinear line segments intersect?

I am writing a function to find the intersection of a pair of lines and another function to find the intersection of a pair of line segments. The parallel case and the single point intersection case ...
0
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1answer
313 views

A Golden Ratio Symphony! Why so many golden ratios in a relatively simple golden ratio construction with square and circle?

Note: this construction is a vastly expanded version of my earlier construction here: Have you seen this golden ratio construction before? Three squares (or just two) and circle. Geogebra gives PHI or ...