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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Prove that if $l$ is a line in the classical euclidean plane, then there is a point $p$ that lies on $l$

Suppose that $\mathbb{P}$ is a Classical Euclidean Plane (satisfies all five of Euclid's postulates). Can you prove that if $L$ is a line in $\mathbb{P}$, then there is at least one point $p$ in ...
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116 views

Crossed Ladders Problem

Two ladders, one 10 meters long and the other 8 meters [long], have been placed in a trench as indicated in the opposite figure. Their point of intersection, M, is 3 meters from the base of the ...
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2answers
43 views

How can I use Menelaus' theorem here (Simson line)?

Given 4 points on a circle A, B, C, and P. Draw the orthogonal projections of P onto triangle ABC and call them $P_1, P_2,P_3$. Show that $P_1, P_2,P_3$ are collinear. After drawing this out, I ...
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1answer
45 views

How would I use vectors for this geometry problem?

Consider a quadrilateral ABCD. K, L, M, N are the midpoints of the segments AB, BC, CD, DA respectively. O is the intersection point of LN, KM. Let P and Q be the middle points of the diagonals AC ...
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2answers
62 views

Trisecting the sides of a triangle.

Consider the hexagon formed by the six points which trisect the sides of a triangle(two on each side). Is is true that when we connect opposite points in this hexagon, the lines intersect at a single ...
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35 views

Intercept planet following an elliptical path (i.e. interplanetary space travel)

So, just as in this question (Intercept path to object following an elliptical path) I have a simple game where I want spaceships to intercept planets, which follow elliptical paths (in my case ...
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3answers
43 views

Find the area of the triangle under certain preconditions

With vertices $(0, 0)$, $(b, a)$, $(x, y)$, prove the area of this triangle is $\frac{|by - ax|}{2}$. We know area of a triangle = $\frac{rh}{2}$. ($r$ is the base.) Well, we have $r =$ the ...
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0answers
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$\frac{4}{3}(1,-1,0)^T+\textrm{vect} \left((-1,1,-1)^T\right)$

Here I have a passage on the conclusion, but I do not understand it. the conclusion say : $f$ is the oriented axis of screw.. how I can from $\frac{4}{3} \begin{pmatrix}1\\1\\0 ...
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1answer
25 views

Getting the intersection of a line and a plain

My line (2,1,10) goes through the plain with the normal (-2,3,8). Now I would like to calculate the intersection with following ...
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2answers
20 views

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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35 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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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
19 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
40 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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1answer
38 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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1answer
55 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
53 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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1answer
28 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
18 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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2answers
43 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
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
26 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
64 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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1answer
54 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
157 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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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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1answer
34 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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0answers
11 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
79 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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1answer
52 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
32 views

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
40 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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3answers
305 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
75 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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0answers
24 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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0answers
30 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 ...
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1answer
106 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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1answer
44 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 ...
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3answers
87 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
33 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
93 views

Why are these three lines concurrent?

Consider a triangle $ABC$ with incentre $I$ and let $AI \cap BC=D$. Let the incentres of $\triangle ACD$ and $\triangle ABD$ be $E$ and $F$ respectively. Prove that $AD$, $BE$ and $CF$ are ...
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0answers
62 views

Polygons with coincident area and perimeter centroids

Let $P$ be a simple, planar polygon. Define $c_a$ as the area centroid of $P$, i.e., the center of gravity of the closed shape $P$. Define $c_p$ as the perimeter centroid of $P$, the center of gravity ...
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1answer
26 views

Comparison of the Chebyshev centers and radii of a set and of its bounding box

The Chebyshev center of a bounded set $Q$ having non-empty interior is defined in this question as the center of the minimal-radius ball enclosing the entire set $Q$. Let $B$ be the minimum-volume ...
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1answer
26 views

Rotation of curve function

I am working on some coding where I require expertise in field of Mathematics. I have a function: $$F(x) = -0.007x^4 + 1.971x^3 - 190.4x^2 + 8150x - 13024$$ I want to rotate some section of this ...
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0answers
35 views

Estimating distance between two points in n-dimensional space, with knowledge of other paths

Suppose there exist four randomly distributed points in $n$-dimensional space: $A$, $B$, $C$, and $D$. We have no knowledge of the coordinates of any of these points, but we do know nearly all of the ...
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1answer
65 views

What's the exact definition of symmetry?

My question is basically the same as this, but I haven't found the answer given satisfying. The definition of symmetry I've come up with is this: Let $X \subseteq R^2$. A symmetry of $X$ is an ...
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1answer
57 views

Can Point inside quad be determined with angles alone?

Given A Quad($C$, $D$, $E$, $B$) and Points $A$, $G$, $F$ Question Is it possible by calculating the angles between points to determine whether a point is inside (including on), or outside the quad ...
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1answer
25 views

Inner product respect on a non-canonical base

Let a,b be vectors, on the standard base we use the dot product by simply doing a.b. But when we consider an other base we put a symmetric matrix between them. Why? How does that work? Thanks
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1answer
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Does this operation result in a convex set

Denote by $cm(B)$ the center of mass of the set $B\subseteq\mathbb{R}^2$. Given two convex sets $A,X\in \mathbb{R}^2$, define $Y$ in such a way that $X\cap A\neq \emptyset$ if and only if $cm(A)\in ...
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1answer
76 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 ...