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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Maximum value of the area of triangle

If two of the medians of a triangle have lengths x and y, what is the maximum value for the area?
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2answers
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Why $(h,k)$ in equation $y= a(x-h)^2 +k$ is the vertex of a parabola?

As in the title , I know how to convert normal explicit equation to a vertex form equation by completing the square . But what is the reasoning behind why $(h,k)$ must be the vertex , but not other ...
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14 views

midpoint of the diagonal of the quadrilateral and rhombus

$EBA,FCB,GDC,HAD$ is a similar triangle which is drawn externally of quadrilateral $ABCD$, where the sides of quadrilateral $ABCD$ become the base of the similar triangle. Let $M,N,P,Q$ are midpoints ...
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3answers
359 views

Euclidean geometry exercise

I would like some help to solve this: Consider a triangle $\triangle ABC$ with $\angle A$ a right angle and $BC=20$. Divide $BC$ into four congruent segments, that is, take the points ...
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7answers
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Why Do The Axioms of Euclidean Geometry Not Need To Include the Definition of Space?

EDIT: update, I found that Euclid's axioms are not considered rigorous. David Hilbert did a full axiomatization of Euclidean Geometry (1899 in his book Grundlagen der Geometrie--tr. The ...
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1answer
13 views

Orthogonal Coordinates

I'm hoping someone could give me a good definition of "orthogonal coordinates." Attempts to find one online has left me only with a vague idea. A reference text would be appreciated.
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1answer
734 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 where ...
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2answers
47 views

How to smooth a list of angles.

I'm not a math guy so maybe there is a super simple thing that my eyes cannot see. And sorry if my math terminology is not good at all. Please address me the right math terminology to use because ...
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1answer
13 views

Cyclic quadrilateral problem

In convex quadrilateral $ABCD$, $AB=2$, $AD=4$, and $2BC+CD=10$. If angle $DAC$ equals angle $DBC$, and the diagonals of $ABCD$ are perpindicular to each other, what is the area of $ABCD$? I have a ...
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24 views

Given $4$ points in the space, how do you check if an arbitrary point is within the area marked by those points?

Given $4$ arbitrary points in the space $A(x_1,y_1), B(x_2,y_2), C(x_3,y_3,), D(x_4,y_4)$, how do you check if an arbitrary point $X(x_5,y_5)$, is within the quadratic area marked by the $4$ points ...
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34 views

Archimedean Arbelos: seven circles with many properties [on hold]

On a line $l$ there exists $3$ points $A,B,C$ where $B$ is located between $A$ and $C$. Let $ \Gamma1, \Gamma2 ,\Gamma3 $ be circles with $AC,AB,BC$ as diameter respectively. $BD$ is a segment, ...
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143 views

Why exactly is Bourbaki difficult? [closed]

I keep hearing people say that Bourbaki is difficult for most undergraduates but I still don't understand why. Surely if it starts from definitions/axioms then practically anyone should be able to ...
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1answer
71 views

Pentagon Forms a 10-sided Polygon Ratio Problem

Let $A_1A_2A_3A_4A_5$ be a regular pentagon with side length $1$. The sides of the pentagon are extended to form the $10$-sided polygon shown in bold in the picture that I have attached. Find the ...
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2answers
38 views

$ABCD$, $P$ is any interior point, $PA=24, PB=32, PC=28, PD=45$

could anyone tell me how to solve it? I have a convex quadrilateral $ABCD$, $P$ is any interior point, $PA=24, PB=32, PC=28, PD=45$ cm, I need to know the perimeter of $ABCD$. Thanks for helping. ...
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0answers
36 views

Triangle $ABC$ and equilateral triangles $ABC'$, $BCA'$ and $ACB'$.

We consider a triangle $ABC$ whose angles are less then $120°$ and construct the equilateral triangles $ABC'$, $BCA'$ and $ACB'$, exterior to $ABC$. $I$ denotes the intersection of $(AA')$ and ...
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3answers
42 views

Given an equilateral triangle, show that $MA + MC = MB$.

I have to solve the following problem: Consider an equilateral triangle $ABC$ and $\mathcal{C}$ its circumscribed circle. Let $M$ be a point located on the arc of the circle defined by $[AC]$ which ...
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0answers
19 views

Study the caracteristics of the transformation $f=r\circ t \circ h$.

Let $OABC$ be a square with $(\vec{OA},\vec{OC})=\frac{\pi}{2}$. Let $r$ be the rotation of center $B$ and angle $\alpha=\frac{\pi}{2}$, $t$ the translation of vector $\vec{CA}$, $h$ the homothetic ...
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1answer
23 views

Similarity of triangles?

The question is: "$ABCD$ is a quadrilateral in which angle $B =$ angle $C$ and $AC$ bisects angle $BAD$. If $BA$ and $CD$, when extended, meet at $E$, prove that $AD/DC = AE/BE$." I'm finding this ...
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1answer
17 views

Determining direction from three points on a line

I have a small geometry problem that for some reason I just can't get a grasp on. You're given three points on a line in 3D space, p1, p2, p3. (assume for simplicity that they're named ...
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29 views

equal length at the tangential [closed]

Let $\Gamma$ is a circle with diameter AB.Let $l$ be the tangent of $\Gamma$ at $A$ ,and $m$ be the tangent of $\Gamma$ through $B$.Let $C$ be a point on $l$ different from $A$,and let $q_1$ and $q_2$ ...
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0answers
12 views

Curves with a property about intersecting hyperplanes

I would like examples of curves in $\mathbb{R}^n$ with the property that any hyperplane of n-1 dimension through the origin intersects the curve at $\leq$ n points. In $\mathbb{R}$, the circle with ...
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1answer
24 views

How many points $P$ such that $\angle APB=\angle BPC=\angle CPA $ are there?

Given that $\triangle ABC$ is arbitrary. How many points $P$ such that $\angle APB=\angle BPC=\angle CPA $ are there?
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1answer
41 views

Why ternary diagrams work

I am trying to understand why ternary diagrams work. In order that the altitude criterion be valid, if I correctly understand, given equilateral triangle $ABC$, whose vertices I name as the three ...
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34 views

Necessary and sufficient condition for Euclidean geometry to hold? [on hold]

If a space obeys the axioms set out by Euclid in 'The elements' is this a necessary and sufficient condition for the geometry in that space to be Euclidean?
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1answer
42 views

“Polysticks” in 3d

Consider a finite set of three-dimensional Euclidean vectors with integer components. How many three-dimensional closed loops can I construct with them? How many of them are elementary, i.e., cannot ...
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1answer
46 views

Volume of the intersection of two tetrahedra

First, I am far from a mathematician, and this question may be easy, if that's the case, please don't hesitate to let me know. Suppose I have 2 tetrahedra (2 3D simplex), with known ...
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1answer
143 views

euclidean geometry books…

I consider myself poor in plane euclidean geometry. so I need a good geometry book which contains very good theory, and a collection a large number of solved problems, and the end of each part.This ...
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25 views

Is there any algorithm for finding the minimum distance to the complement of a convex set?

There have been some algorithms for finding the projection from a given point onto a convex set. This problem seems to be quit easy because of the convexity of the set. However, in the case of finding ...
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1answer
92 views

Is there a name for the recursive incenter of the contact triangle?

Recently, I became aware that there are many more triangle centers than the four I learned about in school. This reminded me of a thought I had when I first learned about the incenter: what point ...
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10 views

Dual Objects and Symmetries

In the study of symmetries of platonic solids (tetrahedron, cube, octahedron, ..), I came across the following. Group of rotational symmetries of a cube is $S_4$. Since octahedron is dual to cube, ...
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21 views

Mathematics-Oriented 4-D Glossary?

Is there somewhere a comprehensive glossary of words or phrases describing geometric concepts or objects in the Euclidean (not Einsteinian) fourth dimension? I have seen a glossary which purported to ...
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1answer
30 views

Euclidean problem of geometry

Let the two quadrilaterals ABCD and EFGH been given: Let's take these hypothesis: $AD = EH$ $A\hat{B}D=A\hat{C}D=E\hat{F}H=E\hat{G}H$ $AC=EG$ The triangle ABD is isosceles and equal to the ...
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6 views

Intersection/union of hyperballs in Minkowski space

I'm trying to manipulate hyperballs bounded by the Minkowski distance. What I would like to do is take the intersection/union of two hyperballs and then find the smallest hyperball which covers the ...
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1answer
39 views

calculate points coordinates on plane from their distances matrix

Given a list of points on a plane is simple to generate a distances list between each pair of points. Pseudo Code: ...
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35 views

Cyclic convex quadrilateral property

Let $ABCD$ be a cyclic convex quadrilateral, and let $P$ be the intersection of the diagonals. Show that $\frac{PB}{PD}=\frac{AB}{AD}\cdot \frac{CB}{CD}$. I guess I need to use Ptolemy's ...
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3answers
202 views

Scaling and rotating a square so that it is inscribed in the original square

I have a square with a side length of 100 cm. I then want to rotate a square clockwise by ten degrees so that it is scaled and contained inside the existing square. The image below is what I'm ...
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1answer
31 views

What is the fundamental theorem on discrete groups of Euclidean spaces?

I have been reading the book Using Algebraic Geometry by David A. Cox, John Little, Donal O'Shea for a university project. I am not clear as to what exactly in meant by the phrase "the fundamental ...
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1answer
367 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 ...
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3answers
53 views

Every reflection is an isometry proof

The theorem is that every reflection $R_{S}$ in an affine subspace $S$ of $\mathbb{E}^{n}$ is an isometry: $R_S:\ \mathbb{E}^{n} \rightarrow \mathbb{E}^{n}:\ x \mapsto R_{S}(x) = x + 2 ...
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1answer
74 views

At most $2n$ vectors, the angle between which $\geq\pi/2$.

In a previous question it is proved that in $\mathbb R^n$ there are at most $n+1$ vectors, the angle between which $>π/2$. How to prove that there are at most $2n$ vectors, the angle between which ...
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1answer
43 views

Alternative word for Euclidean Geometry

If Euclid has only collected the geometry stuffs while books of the other geometer have been burnt, calling the main branch of geometry under name of him might look academically unethical for some ...
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1answer
25 views

If three circles have two common points, prove that every circle that is orthogonal to two circles is also orthogonal to third.

Three circles are given $k_1$,$k_2$,$k_3$ that have two common points A and B. Prove that every circle $k$ that is orthogonal to circles $k_1$,$k_2$, is also orthogonal to $k_3$. Here is my proof ...
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0answers
11 views

Contructions of a rectangular hyperbolar and a inscribed parabola [on hold]

Please see: http://mathworld.wolfram.com/ChaslessPolarTriangleTheorem.html http://forumgeom.fau.edu/FG2004volume4/FG200427.pdf I proposed problem construction of a rectangular hyperbolar and a ...
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ABC is a triangle, D is a point in the triangle. E is the midpoint of BD. AB=BC, angle ABD= angle DBC=35 degrees, angle ACD=25 degrees. Angle BAE=?

I tried to solve this problem but couldn't. I just know that here, angle BDC= 100 degrees, angle BAC= 40 degrees, AB^2+AD^2=2(AE^2+BE^2) and AB/AD={sin(angle DAE)}/{sin(angle BAE)}
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1answer
41 views

Determine Euler Angles from look, up, and cross vectors

I have a spaceship flying through a $3D$ space. The flight is determined by applying a quaternion to the look, up, and cross vectors with the following scheme (this is working perfectly): starting ...
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1answer
71 views

At most $n+1$ vectors, the angle between which $>\pi/2$.

In a $n$ dimensional Euclidean space $V$, there exists at most $n+1$ vectors, each pair has inner product $<0$. This is geometrically obvious in $3$ dimensions...But how can we prove it ...
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27 views

On the congruence of triangles

Sorry for the perhaps somewhat trivial question, but are the criteria for the congruence of two triangles, i.e. "side-angle-side", "side-side-side" and "angle-side-angle", taken as postulates or can ...
8
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1answer
128 views

Solve $10x+2x^2+x^3=20$ using only algebra and geometry?

The cubic formula and modern math is not allowed, only algebra, geometry, and the like. Supposedly this problem was given to Fibonacci. Here is the whole paragraph I read: In Flos Fibonacci gives ...
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1answer
14 views

What is the proper name of a point a long a smooth curve where the radius changes but not direction of curvature?

What you call a point a long a smooth curve where the radius changes? When it reverses curvature, it’s an “inflection point”. What if it doesn’t change direction, just radius? I seem to remember ...
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1answer
41 views

Constructing the inverse of a number geometrically.

this picture: shows a way to construct the inverse of a number $a\ge1$. but how can we construct for a number that is less than 1? My try:: Q1: is my try correct? Q2: how to prove ...