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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553 views

What is a the intuition behind a parametric equation?

I have always used equations for the line (y=a + bx) in R2. Recently I came upon this thing called parametric equations. I cannot grasp the difference between them and the equations for lines that I ...
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4answers
3k views

How do I move through an arc between two specific points?

I've found many answers to similar questions here, but I'm still stuck. I want to move an object from point sx,sy to point dx,dy through an arc that bulges by distance b from the line straight ...
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1answer
54 views

Geometry finding area problem

A regular 2N -sided polygon of perimeter L has its vertices lying on a circle. Find the radius of the circle and the area of the polygon.
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0answers
70 views

About the relation between a tetrahedra and spheres moving in a tetrahedra

I found the following question in a book: There exists a regular triangle $OAB$ which has edge-length $2$. Let $H, I, J$ be a foot of the perpendicular line drawn from a point $P$ in $OAB$ to the ...
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3answers
512 views

Intuition - The Shortest Curve Between Two Points is a Line

The references below aver that the following is not crudely trivial: The shortest curve between two points is a (straight) line. An elementary school teacher construed it as follows ...
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1answer
116 views

Find the volume of the region where a line segment can move

I and my friends enjoy making and solving new mathematical questions. We made the following question, but we are facing difficulty. Could you show me how to solve this? Question: Let $a, b$ be ...
105
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3answers
14k views

Slice of pizza with no crust

The following question came up at a conference and a solution took a while to find. Puzzle. Find a way of cutting a pizza into finitely many congruent pieces such that at least one piece of pizza ...
5
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1answer
542 views

Straightedge-only construction of a perpendicular

There is a circle in the plane with a drawn diameter. Given a point inside the circle (not on the diameter or the circle), draw the perpendicular from the point to the diameter using only a ...
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1answer
124 views

Angle between the orthocenter and incenter

From I am asked to get the value of $\theta$ in terms of $\alpha$ and $\beta$. Quick geogebra "shows" it is the difference, but how do you show it mathematically? All the properties I have about it ...
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1answer
258 views

Triangle inequality question.

For triangle ABC, there is a point X such that B-A-X. There is any point P on bisector of exterior angle CAX. Prove that PB+PC>AB+AC. I have tried for many hours, but i have no idea how to solve it. ...
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2answers
2k views

Parallel postulate from Playfair's axiom

Parallel postulate: If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, ...
2
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1answer
104 views

Plane isometries $g\ , f$ , properties of fixed points and types

Given two plane isometries $g\ , f$ and $f^{'} = g\circ f \circ g^{-1}$ prove that: If $P$ is a fixed point of $f$ then $g\left(P\right)$ is a fixed point of $f^{'}$ and if $Q$ is a fixed point of ...
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1answer
318 views

Chord passing through concentric circles.

A chord $AB$ of one of two concentric circles at intersect each other at $C$ and $D$. We have to prove, $AC=BD$. I am not sure what this question means by 'intersect each other', but if I am ...
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votes
2answers
146 views

Find The range of $r/R$.

Given a triangle $ABC$ with angle $A=90^{\circ}$. Let $M$ be the midpoint of $BC$. If the inradii of the triangles $ABM$ and $ACM$ are $r$ and$\ R$ respectively, then find the range of $\dfrac rR$ .
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1answer
54 views

Can convex hulls contain duplicate points?

Given 4 vertices: (-3.2, 0.8), (-3.2, -0.8), (3.2, -0.8), (3.2, 0.8) A function which I did not write, and given vertices, will return the points of a convex hull. Using the points above, it ...
3
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2answers
141 views

Obvious statement, but how do you formally prove it?

Let d be a line, M a point on the line, and n a positive integer. Why is there exactly two points at n distance from M on d? How to prove it with Euclidian axioms (without algebra) ?
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3answers
549 views

Alternative model of Euclidean geometry

I'm planning to teach high-school geometry. As usual, this will be by building from axioms. (The axioms used are AFAICT particular to the book I've been assigned, but they're some combination of ...
2
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2answers
433 views

Classification of the orientation preserving isometries of $\mathbb{E}^4$.

I can't find a classification of the orientation preserving isometries of $\mathbb{E}^4$. I found this e somewhere on the internet but of course it also raises the question of there being a general ...
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0answers
58 views

Determine vectors similarity weighing the vectors lengths

I have many vector couples, for which I would like to assign a single number (for each couple) that signify their similarity. I am running a minimization algorithm on the whole group of vector ...
3
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1answer
281 views

Could someone explain this animated gif to me in mathematical terms?

I understand that the area of the two squares around the right triangle are the total area of the one that is the hypotenuse. Is this just a proof for the Pythagorean theorem or is there some other ...
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1answer
372 views

A problem on triangle and its perpendicular bisectors.

I'm trying to solve the following problem : "In △ABC, coordinates of $B$ are $(−3, 3)$. Equation of the perpendicular bisector of side $AB$ is $2x + y − 7 = 0$. Equation of the perpendicular ...
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2answers
87 views

Optimal Configuration for a Set of Points

Consider a set of $n$ points on the plane with positions $\mathbf{p}_1,\dots,\mathbf{p}_n$, such that each point $i$ has at least one neighbor $j$ at a distance of no more than $\lambda$ away from it ...
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2answers
262 views

How well can we embed graphs with shortest path metric into $\mathbb{R}^2$ with Euclidean metric?

If we take the integer lattice in $\mathbb{R}^2$ and make edges from $(m,n)$ to $(m+1,n)$ and $(m,n+1)$, you get your typical city block street layout, and if we put the shortest path metric on the ...
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4answers
1k views

Finding an invisible circle by drawing another line

A friend of mine taught me the following question. He said he found it in a book a few years ago. Though I've tried to solve it, I'm facing difficulty. Question: You know on a plane there is an ...
2
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2answers
363 views

Converse of the British Flag Theorem

Exist a theorem known as British Flag Theorem. It say that in a rectangle $ABCD$ we have $PA^2 - PB^2 + PC^2 - PD^2 = 0$, for any point $P$ in the plane. I was thinking in a type of converse of this ...
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0answers
244 views

euclidean distance vs squared euclidean distances in 1 dimension, which one is the best?

I would like to compare the "disparities" between two groups. I define the mean disparity of a group as the mean of all the euclidean distances within the group with both groups presenting their ...
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2answers
203 views

Euclidean geometry prerequisites

I have used enrolled in a introduction to Euclidean geometry course, but I have very little experience with geometry, almost none. I have an engineering background so I have taken calculus, linear ...
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2answers
442 views

Does the line connecting the mid-points of two opposite sides of a quadrilateral divide it equally?

Suppose ABCD is a general quadrilateral. P and Q are the mid points of AB and CD respectively. Now, will PQ line divide ABCD in two equal quadrilateral? That is, will the area of APQD and BPQC be ...
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1answer
24 views

How do I find out the coordinates, interpolating across an angled line?

Suppose I know the coordinates of $A$ and $B$. The angle $X$ does not mean the total angle between the red lines, but rather how far along the angle that the purple line is. What is the easiest way ...
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3answers
2k views

A generalization of IMO 1983 problem 6

Note: This question has a bounty that will expire in just a few days. Let $a,b,c$ and $d$ be the lengths of the sides of a quadrilateral. Show that $$ab^2(b-c)+bc^2(c-d)+cd^2(d-a)+da^2(a-b)\ge 0$$ ...
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1answer
1k views

Parametric Equations of an Oblique Circular Cone

I am trying to determine the parametric equations for a specific shape of an oblique circular cone with no success. Exhaustive web searchs and many texts have not been fruitful as regards ...
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0answers
237 views

References for problems related to uniformly distributed points/ arcs on circle

I am looking for references to problems related to computing the statistical properties of uniformly randomly chosen points or arcs on the unit circle, e.g.: On a unit circle, $n$ points are ...
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1answer
73 views

Existance and uniqueness of solution for a point with fixed distances to three other points

I have two sets of known points in $\mathbb{R}^2$: Four points $\mathbf{p_1}, \mathbf{p_2}, \mathbf{p_3}, \mathbf{p_x}$ , and three other points $\mathbf{q_1}, \mathbf{q_2}, \mathbf{q_3}$. I would ...
0
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1answer
69 views

Geometry Problem on Finding Angles

In an isosceles triangle ABC,AB=AC,P and Q are points on AC and AB respectively such that CB=BP=PQ=QA.Then prove that angle AQP=900/7
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3answers
383 views

Geometry problem on circles from a competition

Triangle $\triangle ABC$ is an equilateral triangle whose side is $16$. A circle meets the sides of the triangle at $6$ points: it intersects $AC$ at $G$ and $F$ and $|AG|=2$, $|GF|=13$, $|FC|=1$. ...
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1answer
81 views

Help on whether a geometry solution is valid.

Let $ABC$ and $AB'C'$ be similar right angled triangles with right angles at $C$ and $C'$, respectively. Let $l$ be the line between $C$ and $C'$, and let $D$ and $D'$ be the points on $l$ such that ...
4
votes
1answer
178 views

Euclidean Ramsey theory problem

Let $k\geq 1$ be given. Consider the following statement: For all (non equilateral) triangles (represented by 3 points in $\mathbb R^2$) and for all $k$-colorings of $\mathbb R^2$ there exists a ...
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2answers
60 views

Proving angle equality in a circle

Let $ABDC$ be a quadrilateral inscribed in a circle with centre $O$. If the diagonals $AD$ and $BC$ intersect at point $E$, then we have to prove that sum of angles $AOB$ and $COD$ is equal to twice ...
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2answers
218 views

Why are $\triangle ABC$ and $\triangle CDE$ similar?

Português: Porque $\triangle ABC$ e $\triangle CDE$ são similares na imagem abaixo? Sendo que $M$ e $N$ são pontos médios de seus segmentos. $AD$ e $BE$ são alturas relativas. $$$$ English: Why are ...
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1answer
236 views

A geometry problem on locus

Prove that the locus of the midpoints of the parallel chords of a circle is the diameter of the given circle.
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2answers
142 views

A plane geometry tough problem

$ABCD$ is a quadrilateral. $P,Q,R,S$ are the midpoints of $AB,BC,CD,DA$ respectively. $PR$ and $SQ$ intersect at $L$. $T$ is any point within the quadrilateral. Prove that ...
2
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0answers
89 views

quadrature formula for a lune

I have been reading about quadrature formulas in the complex plane. On the set $\mathbb{D} = \{|z| < 1\}$ we have $\int_{\mathbb{D}} f(z) dA = \pi f(0)$. Standard result in Harmonic functions. ...
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0answers
665 views

Minimal number of moves to construct the challenges (circle packings and regular polygons) in Ancient Greek Geometry?

In the web game Ancient Greek Geometry, there are challenges to construct regular polygons and circle packings using ruler and compass constructions. The game measures the number of line and circles ...
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1answer
174 views

Triangle inequality for an obtuse triangle

$\alpha < 45^\circ$, how to show that 1) $|AB+AC|>|DB+DC|$? 2) $|AB+AC|>|DB+DC+DA|$?
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0answers
44 views

light geometry and art

I'm a professional artist. I have a question regarding light logic. I have simplified the problem a bit. Imagine a segment in 3D space, AB of known length and a point light illuminating the ...
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2answers
354 views

Area of quadrilateral inside a triangle.

Let us consider an isoscles triangle ABC where $AB=AC=\sqrt{13}$,and length of $BC=4$.The altitude on $BC$ from $A$ meets $BC$ at $D$.Let $F$ be the midpoint of $AD$.We extend $BF$ such that it meets ...
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3answers
249 views

how to calculate the area of ​​a rhombus is in $cm^2$

How do I calculate the area of ​​a rhombus is in $cm^2$? Is the formula $\frac12 \times 17 \times 16$? Anyone can help me to solved this? I don't know the rhombus formula. Based on my exercise ...
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0answers
140 views

Bound on the angle between a vector and a subspace

Suppose you have three complex vectors $x_1$, $x_2$, and $x_3$. Define $a = \angle(x_1,x_2)$, $b = \angle(x_1,x_3)$. My question is about $c = \angle(x_1, span(x_2,x_3))$, the angle between the vector ...
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2answers
392 views

Proof of the following: How many $(n-2)$ dimensional faces from a corner of a hypercube

I asked a question earlier regarding the number of $(n-2)$ dimensional faces exiting a corner of an $n$ dimensional hypercube. (For example the number of points in a corner of a square, or the number ...
6
votes
4answers
465 views

Some theorems in euclidean geometry have incomplete proofs

I have seen that, in euclidean geometry, proofs of some theorems use one instance of the 'geometric shape'(on which the theorem is based) to proof the theorem. Like, the proof of 'A straight line ...