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)

3
votes
1answer
39 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 ...
0
votes
1answer
27 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
vote
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
votes
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 $...
0
votes
1answer
77 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
55 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
votes
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 ...
0
votes
0answers
24 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 ...
1
vote
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 ...
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 ...
1
vote
2answers
21 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 ...
0
votes
0answers
32 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) :=...
4
votes
4answers
365 views

Find the sum of angles without trigonometry?

I have found the sum it's $180$ but using right triangle and sine theorem.
1
vote
1answer
21 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
votes
1answer
54 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
votes
1answer
110 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
votes
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 ...
2
votes
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
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 ...
0
votes
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{...
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
votes
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 ...
0
votes
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
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 ...
2
votes
3answers
171 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 $T_2$ ...
1
vote
1answer
77 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
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
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 ...
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
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 ...
0
votes
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,...
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 $\angle AMC$? ...
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, ...
1
vote
1answer
24 views

Victoria Jones want to construct a time capsule. The capsule will be right circular cylinder

Victoria Jones want to construct a time capsule. The capsule will be right circular cylinder of height 'h' cm, and radius 'r' cm on each end . Let the total volume of capsule be V cm^3. Express V in ...
1
vote
0answers
28 views

Convex quadrilateral with interior point

Let $ABCD$ be a convex quadrilateral and $X$ is an interior point. Also let $AX\cap BD=\{E\}$, $BX\cap AC=\{F\}$, $CX\cap BD=\{G\}$ and $DX\cap AC=\{H\}$. Prove that: $$AF\cdot BG\cdot CH\cdot DE=...
4
votes
6answers
636 views

Given distances (shortest paths) between four cities, how to show that they cannot be in the same plane?

In the example below we are given distances between four cities. The author of the book says that these distances "suffice to prove that the world is not flat". Do I understand this correctly that ...
4
votes
0answers
18 views

What makes a geometric construction more or less stable?

As anyone who's actually done geometric construction of n-gons knows, not all construction methods are made equal. Some are very stable (the shape you get is always close to ideal even if you're not ...
0
votes
2answers
28 views

Draw polynomials to demonstrate Euclid's axioms.

I've a problem with Euclid's axioms. I understand them, but now I want some equations (polynomials) that I can use to draw some graphics and probe these axioms. For example, a rect equation that ...
1
vote
1answer
44 views

Geometric proof of expansions of series

I have read that Barrow had proved the fundamental theorem of calculus. I have read that proof and its a good. Further I know Newton had derived the sine and cosine series. His methods obviously didn'...
-1
votes
0answers
12 views

tangent line to the circle at P(0.87,0.50) intersects the X-axis at the point Q(x,0). Find x?

How to find tangent line to the circle at P(0.87,0.50) intersects the X-axis at the point Q(x,0). Find x. What equations can I use to answer this
0
votes
1answer
33 views

Finding points on a right triangle

I have the points A, B and C. I also have the angle alpha between AB and BD or BE, and I know l = |BD| or |BE|. But how can I find D or E?
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 ...
0
votes
0answers
27 views

how to find the angle of Lovasz umbrella

in the book Thirty-three Miniatures: Mathematical and Algorithmic Applications of in problem 28 The Secret Agent and the Umbrella page 132 (pdf 140) we want to find an orthogonal reperesentation of ...
6
votes
2answers
159 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, ...
2
votes
2answers
24 views

Prove that $\frac{AB}{PB}\cdot \frac{BQ}{QC}\cdot \frac{CR}{RD}\cdot\frac{DS}{SA}= 1$

If the sides AB, BC, CD and DA of a quadrilateral ABCD are cut by straight lines at points P, Q, R and S respectively, how do I prove that $\frac{AB}{PB}\cdot \frac{BQ}{QC}\cdot \frac{CR}{RD}\cdot\...
-1
votes
2answers
124 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 ...
1
vote
1answer
49 views

An isometry preserves straight lines

This is an exercise of the book "Basic Mathematics" by Serge Lang, p.145. I've been working on this proof for a few days and I can't seem to make it more coherent than this. Would appreciate some help....
3
votes
0answers
34 views

When does there exist a point with a given ratio of distances to the vertices of a triangle?

I have the triangle ABC and an unknown point P not necessarily inside the triangle. Also, I have three lengths (...