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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53 views

Are there some undiscovered/unproved theorems about Euclidean triangles?

For a particular case of a figure called simplex, a triangle is surprisingly complicated (in my opinion). As an illustration, see the list of triangle topics on Wikipedia, and the Triangle page. The ...
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1answer
30 views

Does it have to be a right angle?

Say you have a circle $O$ and a point on the circle $P$. From P, we create 2 points $A$ and $B$ on the circle such that $PA=X$, $PB=Y$, and the 2 points are on different sides of $\overline{PO}$ (...
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1answer
45 views

Solving this without the concept of similarity?

Any point $X$ is taken on the side $BC$ of $\Delta ABC$. Prove that $AX$ is bisected by the straight line joining the midpoints of $AB$ and $AC$. This problem is trivial when one uses the concept of ...
2
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2answers
55 views

Is there a theorem of intersecting chords in an ellipse?

I found a well known theorem that if $A,B, C$ and $D$ are on the circumference of a circle and $AB\cap CD=P$ then $AP\cdot BP=CP\cdot DP$ . Is there anything generalization of it to an ellipse? Maybe ...
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0answers
25 views

pack equilateral triangle

I'm working on a problem of inscribing equilateral triangle for some time now and it goes like this : using only a foot rule and a compasses , show a way of inscribing an equilateral triangle into ...
6
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1answer
93 views

Proving an exercise from my High School Geometry Class

In my class we are learning geometry and the instructor gave us this problem: Let $ABC$ be a scalene triangle with orthocenter $H$, and let $W$ be a point on the side $BC$, lying strictly between $B$ ...
2
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2answers
33 views

Prove on Incenter and mid point.

Let the incircle (with center $I$) of $\triangle{ABC}$ touch the side $BC$ at $X$, and let $A'$ be the midpoint of this side. Then prove that line $A'I$ (extended) bisects $AX$.
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1answer
62 views

How do I convert $ 2=\sec(θ) $ into rectangular? [closed]

For my homework, I need to convert the polar equation of the "curve" $\sec\theta=2$ into rectangular form. I assume this must be an equation between $x$ and $y$, but the the answer according to my ...
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1answer
32 views

Intuitive explanation of Pascal's Theorem

I am wondering why Pascal's Theorem, as well as other 'Euclidean' results in projective geometry like Brianchon's Theorem should be true for not only circles, but also conics in general. Is there ...
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1answer
140 views

Would Euclid be satisfied by the construction of the 17-gon given by Gauss?

In our lecture on Algebra we were given the following exercice: Construct the regular 5-gon using straightedge and compass. (only using elementary geometric reasonig) If you construct the length ...
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3answers
47 views

What is the relationship between average distance between 2 random points in a cube, with min and max?

I wrote a computer program to simulate the minimum, average, and maximum distance between $2$ random points in a unit length cube (such as $1$ cubic foot). The minimum possible distance is $0$ and ...
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2answers
62 views

Solve using Butterfly Theorem.

Let $PT$ and $PB$ be two tangents to a circle, $AB$ the diameter through $B$, and $TH$ the perpendicular from $T$ to $AB$. Then prove that $AP$ bisects $TH$.
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1answer
18 views

Prove between Simson line & Nine point circle.

Prove that the Simson lines of diametrically opposite points on the circumcircle are perpendicular to each other and meet on the nine-point circle. I proved the first part of the problem but not able ...
0
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1answer
47 views

Circumcenter of triangle

I already know that, for a triangle $\Delta ABC$ $G$ is the triangle centroid, we have $$\vec{GA} + \vec{GB} + \vec{GC} = \vec{0}.$$ $I$ is the incenter, we have $$a\vec{IA} + b\vec{IB} + c\vec{IC} =...
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2answers
70 views

Euclidea 3 9.8 Chord Trisection

Construct a chord of the larger circle through the given point (on circumference of larger circle) that is divided into three equal segments by the smaller circle (circles are concentric) I'm having ...
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1answer
56 views

A new Equation of a Straight Line

Have I actually discovered a new straight line Equation? (y-y2)/(y-y1)=(x-x2)/(x-x1) is also a straight line equation?
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3answers
36 views

Prove for Pedal & Isosceles triangle.

The tangents at two points $B$ and $C$ on a circle meet at $A$. Let $A_1B_1C_1$ be the pedal triangle of the isosceles triangle $ABC$ for an arbitrary point $P$ on the circle, as shown below. Then ...
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1answer
45 views

Necessary & Sufficient condition for the line $ax+by+c=0$ to pass through the 1st quadrant

What is the necessary and sufficient condition for the line $ax+by+c=0$, where $a,b,c$ are non-zero real numbers, to pass through the first quadrant? I could find the points at which the line ...
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0answers
11 views

Split a map into roughly equal sections directionally and put points in it

I have a 16000 x 9000 grid map and I want to split it into x sections that are preferably of equal size. Then I want to place points on each section are centers of circles with a 2200 unit radius and ...
2
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1answer
71 views

“Universal property” of cross product

Let $V$ be a three dimensional euclidian vector space which is oriented. Because of the orientation, we can define the cross product $\times: V^2 \rightarrow V$ uniquely by: $<v\times w,u> = \...
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2answers
40 views

Can a curve be rotated and translated to 'fit inside itself'?

Working in $R^2$, consider any continuous curve $C$ with endpoints $a$ and $b$, such that the curve does not intersect the line formed by $a$ and $b$. Then the line $ab$ and $C$ form an enclosure, $E$....
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3answers
48 views

Finding the length of the sections of this side

Let there be a triangle $ABC$ with angle bisector $AL$ dividing $BC$ into $BL$ and $LC$. Can it be proven that $BL = \dfrac{ac}{b+c}$ and $LC = \dfrac{ab}{b+c}$? Motivation: I'm trying to solve a ...
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0answers
21 views

Barycentric coordinates

Can anybody here explain to me, starting from scratch, how it is that barycentric coordinates work? In case you don't have the time to do so, what's in your opinion the book (or books) that anybody ...
3
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3answers
67 views

Prove: In a Triangle, $II_1 = a\cdot \sec \frac{A}{2}$

Prove that $II_1 = a\cdot \sec \dfrac{A}{2}$. $I$ is center of incircle, $I_1$ is center of excircle. What I did is : Drop $ID \perp AB$, & $I_1F \perp AF$ at $F$ So $ID\parallel I_1F$ $\dfrac{...
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1answer
70 views

Proving Gerretsen's Inequality

Today in class we were shown Gerretsen's inequality: $$16Rr-5r^2\leq s^2 \leq 4R^2+4Rr+3r^2$$ Where $R$, $r$, and $s$ are the circumradius, in radius, and semiperimeter of a triangle. After some ...
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3answers
45 views

The value of the perpendicular on a diagonal in a rectangle

How to calculate the perpendicular on the diagonal of a rectangle with sides $2$ and $\sqrt2$ ? I've already calculated the diagonal by using Pythagorean theorem which is $\sqrt 6$. Then I didn't ...
7
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1answer
130 views

When do $n+2$ points in $\mathbb{R}^n$ lie on a same $(n-1)$-sphere?

When $n=2$, the following results are well-known: Proposition 1. Let $A,B,C,D$ be $4$ distinct points in $\mathbb{R}^2$. They are aligned or cocyclic if and only if: $$\left(\overrightarrow{CA},\...
3
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3answers
261 views

Have historians responded to Raju's critique?

C. K. Raju has made some outrageous criticisms of the traditional take on Euclid in particular and Western history in general. Yes he has a book published on the subject with an apparently respectable ...
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3answers
96 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
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1answer
49 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 ...
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1answer
39 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 ...
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1answer
28 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 ...
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2answers
58 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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1answer
91 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
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2answers
83 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 ...
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1answer
33 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 ...
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0answers
26 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 ...
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0answers
70 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 ...
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1answer
25 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 ...
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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 ...
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4answers
394 views

Find the sum of angles without trigonometry?

I have found the sum it's $180$ but using right triangle and sine theorem.
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1answer
26 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 $...
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1answer
58 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
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1answer
122 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 ...
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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 ...
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0answers
27 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
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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 ...
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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{...
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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 ...
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0answers
48 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 ...