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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Triangles formed by diagonals of trapezoids

$\Delta$ AOB and $\Delta$ DOC should be equal in area. Correct me if I am wrong. Given: Trapezoid ABCD with ratio $\frac{area \Delta AOB}{area\Delta ABD}$ = $\frac{3}{4}$. I am trying to find (1) ...
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3answers
70 views

Manhattan distance vs Euclidean distance

Suppose that for two vectors A and B, we know that their Euclidean distance is less than d. What can I say about their Manhattan distance?
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0answers
14 views

Non-equivalent phrasings of Playfair's Axiom which are in use

For example on ProofWiki Playfair's Axiom is given as Exactly one straight line can be drawn through any point not on a given line parallel to the given straight line in a plane. but for example ...
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1answer
40 views

Simple geometric proof of parallel lines cut by transversals

Three parallel lines a,b and c are cut by transversal ABC. I need to prove that, if $AB = BC$, then $A'B' = B'C'$. I've made this drawing in geogebra. Any idea of what theorem is this? Could you ...
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1answer
30 views

Prove congruent angles have congruent supplements.

Prove congruent angles have congruent supplements. I do not yet have degrees. Could I somehow use the base angles of isosceles triangles are congruent?
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1answer
24 views

Prove vertical angles are congruent.

Prove vertical angles are congruent. I don't yet know degrees. All I know is congruent angles have congruent supplements. Is it too easy to just say that if I have two intersecting lines AC and BD ...
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4answers
332 views

Concentric Equilateral Triangles

I'm currently researching a particular dynamical system that is very geometric in nature. As part of this, I need to prove the following results (the second obviously implies the first). They are ...
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1answer
38 views

Find the inverse of defined operation $\Delta$

We defined the operation $ \Delta $ as $ (a,b) \Delta (c,d) = (ac + \delta bd, ad + bc) $ where $ a, b, c ,d \in \mathbb{Q} $ I have already proven that this operation is both commutative and ...
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2answers
42 views

Can you construct (ruler and compass) a square with an irrational area?

I've heard that when $\pi$ was proved irrational, that squaring the circle was not proved impossible. This lead me to believe that you could construct a square with an irrational area. Is this ...
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1answer
51 views

Prove every segment has a midpoint

Prove every segment has a midpoint. Unfortunately I do not have the definition yet of isosceles triangles. All I have is SSS and SAS. I also do not have right angles. But I do have perpendicular ...
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1answer
35 views

Prove every angle has a bisector.

Prove every angle has a bisector. I have successfully constructed a bisector and justified by construction. Now I need to put it in proof form. However, I technically do not know midpoints and ...
2
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1answer
37 views

How to show parallelism

Problem: Given two non-congruent circles that intersect at two points X and Y. One secant segment passes through X and intersects one circle (C1) at A and the other circle (C2) at B. Another secant ...
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1answer
62 views

Why must there be an infinite number of lines in an absolute geometry?

Why must there be an infinite number of lines in an absolute geometry? I see that there must be an infinite number of points pretty trivially due to the protractor postulate, and there are an infinite ...
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0answers
56 views

Pythagorean rectilinear polygons

Polygons all of whose edges meet at right angles are called rectilinear polygons. I am interested in rectilinear polygons with integer distance between each pair of vertices. Such rectilinear polygons ...
2
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1answer
31 views

Is it possible to determine triangle with prescribed centres (incentre, orthocentre, barycentre etc.)

The centres of a triangle is related to the triangle itself, or in the language of coordinate geometry, their coordinates can be calculated from that of the triangle's vertices. Can we reverse this ...
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1answer
53 views

invariant points of an isometry

If $A$ and $B$ are two invariant points of an isometry $f$, then every points in line $AB$ is an invariant point of $f$.
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2answers
132 views

Is Euclidean Geometry studied at all?

Is there a place for Euclidean geometry in the hearts or minds of any mathematicians? I personally find it to be the most beautiful mathematics I have yet encountered but I see little of it on sites ...
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1answer
54 views

How one can show Gerretsen's inequality?

I read from http://rgmia.org/papers/v6n3/wsh.pdf the following: A triangle with semiperimeter $s$, circumradius $R$ and inradius $r$ satisfies $$16Rr-5r^2\leq s^2\leq 4R^2+4Rr+3r^2.$$ How can I prove ...
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1answer
19 views

Difference between $\mathbb{R}^2$ and $SE(2)$

I would like to have a good explanation of which is the difference between the Euclidean Group $SE(2)$ and the Euclidean space $\mathbb{R}^2$. From what I understood in $SE(2)$ there is also a ...
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1answer
29 views

How can we minimize this distance?

Given $A = (p, q)$ and $C = (−q, p)$ a pair of points in $\mathbb{R}^2$. Assume that $q > p > 0$. Find $x, y ∈ R$ such that for $ B = (x, 0), D = (0, y)$, $S = AB + BC − |CD − DA|$ is the ...
7
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6answers
241 views

Is $540^\circ$ a straight angle?

The usual definition of a straight angle is a $180^\circ$ angle. however, because a $540^\circ$ angle is also the same shape, is it a straight angle as well?
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1answer
31 views

How should we minimize the area of this triangle?

Consider the angle between two rays $l_1$ and $l_2$ with a vertex O and point A in this angle. Now Consider all possible triangles with vertex O such that two sides of them belong to $l_1$ and $l_2$ ...
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1answer
34 views

Is it true that these angles are equal?

Suppose we have a line $l$ and points $A$ and $B$ which are on different sides of $l$. Point $P$ is on line $l$. When we maximize $|PA-PB|$, it seems that the angle formed by $PA$ an $l$ is equal to ...
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1answer
19 views

Value of X for which both tha angles are equal?

${C}$ and ${D}$ are two points on the same side of a straight line ${AB}$. Find a point X on AB such that the angles ${CXA}$ and ${DXB}$ are equal. Note: This is how I have approached the problem. We ...
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1answer
40 views

Perimeter of triangle formed by connecting intersection points of altitudes.

Given acute triangle $ABC$ with altitudes $AA_1, BB_1,CC_1$. How do we show that the perimeter of triangle $A_1B_1C_1$ is less than twice the length of any of the altitudes?
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2answers
103 views

How do I deal with reflections inside an ellipse?

Suppose I have an ellipse with foci $F_1$ and $F_2$. How do I show that any ray of light which intersects the segment connecting the foci will have subsequent reflections that always are tangent to ...
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1answer
37 views

A level Ellipse question

An ellipse has the equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, where $a>b$, and with eccentricity $e$. It also has foci $S$ and $S'$ and directrices $l$ and $l'$. a) Use the focus-directrix ...
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1answer
42 views

How do I find the area of a triangle formed by cevians?

Given $\triangle ABC$, locate points $A_1$, $B_1$, $C_1$ on respective sides $BC$, $CA$, $AB$ such that $$\frac{BA_1}{A_1C} =\frac{CB_1}{B_1A} = \frac{AC_1}{C_1B} = 2$$ How can I show that the ...
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1answer
38 views

Prove that for any two points $A$ and $B$ $\overrightarrow{AB} \cup \overrightarrow{BA} = \overleftrightarrow{[AB]}$

Question: Prove that for any two points $A$ and $B$ $\overrightarrow{AB} \cup \overrightarrow{BA} = \overleftrightarrow{[AB]}$ The right hand side of the statement that I am trying to prove is a line ...
0
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1answer
31 views

What can we say about the areas of these two triangles?

Given triangle ABC. Let X be a point on AB, Y be a point on BC and Z be a point on AC. Now suppose we reflect X, Y, Z around the midpoint of the sides they are on and label the images X', Y' and Z'. ...
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1answer
44 views

Projective Geometry with Clifford Algebra - lost Inner Product

Projective geometry may be studied with the tools of Clifford Algebras by adding a new direction (see for example this article). But as far as I understand it, only blades and null spaces are used for ...
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1answer
25 views

Models of Incidence Geometry

First I'd like to thank you for reading this, there's a lot here. Secondly I am having a lot of trouble negating the first axiom which I think is hindering my ability to apply it to these types of ...
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3answers
54 views

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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0answers
110 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
36 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
49 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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0answers
31 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 ...
2
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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
10 views

$\frac{4}{3}(1,-1,0)^T+\textrm{vect} \left((-1,1,-1)^T\right)$

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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0answers
31 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
37 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
36 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
54 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 ...
2
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3answers
51 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 ...