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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44 views

Orthogonality of vectors and its dependence on the inner product.

Consider a set of vectors, $\{{\bf e}_i\}$ in $\mathbb{R}^n$. I am thinking specifically of the standard orthonormal basis. I am having a very difficult time understanding what it means for vectors to ...
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1answer
55 views

Rotation Arc Length in 4D

If I have a point $(x_0,y_0,z_0,w_0) \in \mathbb{R}^4$ and I rotate it within the $xy$-plane ($0<\alpha<\pi$) and $zw$-plane ($0<\beta<\pi$), how can I determine the length of the arc ...
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22 views

Using Affine Transformation to prove Concurrency

Let $ABCDE$ be a convex pentagon with $F=BC\cap DE, G=CD\cap EA, H=DE\cap AB, I=EA\cap BC, J=AB\cap CD$, Suppose that the areas of $\triangle AHI, \triangle BIJ, \triangle CJF, \triangle DFG, ...
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1answer
14 views

Similiar triangle inside a triangle based off of segment from feet of altitudes.

Given some triangle ABC, with feet of altitudes D, E, and F, I need to show that triangle ABC is similiar to triangle AEF. This is an image I made for it I have been able to show that the angles of ...
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1answer
27 views

Shortest expression for the diagonal in a convex non-cyclic quadrilateral knowing its sides and the other diagonal?

I'm trying to arrive to the shortest expression possible for finding the diagonal in a convex (and non-cyclic) quadrilateral, knowing its four sides lengths and the other diagonal. My best try ...
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3answers
58 views

Shown $p \in \mathbb{Z} [i]$ is a prime given $p\in \mathbb{Z}$ is a prime and $p$ does not equal $x^2 + y^2$

Suppose $p \in \mathbb{Z}$ is a prime number for which there are no integers, $x$, $y$ such that $p = x^2 +y^2$. How can I go about showing that $p$ is a prime element of $\mathbb{Z} [i]$. Assuming ...
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2answers
60 views

Find angles of triangle formed by images of vertices about opposite sides of an isosceles triangle

In $\triangle ABC$, $AB=AC$ and $\measuredangle BAC=30^\circ$. If $A^\prime$, $B^\prime$ and $C^\prime$ are the reflections of $A$, $B$, and $C$ about $BC$, $CA$ and $AB$, How to find $\measuredangle ...
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1answer
42 views

Without using angle measure, how do I prove that vertical angles are congruent?

Assume that X is a point between A and C, that X is also between B and D, and that these points are not all collinear. Then the angles AXB and CXD are called vertical angles. Prove that vertical ...
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1answer
18 views

minimization of sum of distances using elementary methods

I want to find a solution to the following minimization problem using only elementary methods. This is to say: high school algebra, basic inequalities, basic trigonometry and trigonometrical ...
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1answer
41 views

Geometry with circles.

Two circles, with centres O and P respectively, intersect at A and B. The extension of OB intersects the second sircle at C and the extension of PB intersects the first circle at D. A line through B ...
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1answer
39 views

Distance between points at circles

May be someone can help me to solve the problem. There are circle with radius R1 and circle with radius R2. We also know the distance between A and O and that angle AOB = $\phi$. The aim is to ...
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2answers
776 views

Area of Intersection of Circle and Square

Given a point $(x,y)\in [0,1]^2$ and $r > 0$, I would like to derive a general formula for the area of the intersection of the circle of radius $r$ centered at $(x,y)$ and the unit square. What is ...
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2answers
23 views

Given $u,v\in \mathbb{S}^n$, then there exists a orthogonal matrix s.t. $u=O v$.

Is the following intuitive statement true: For given $u,v \in \mathbb{S}^n$, there exists a matrix $R\in O(n)$ s.t. $u=Rv$.
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1answer
17 views

Prove/disprove statement about functions with special properties

Suppose $H:\mathbb{R}^{2n}\to\mathbb{R}$ satisfies the following: $H\geq0$ everywhere and $H(x)=0\iff x=0$; $H\in\mathcal{C}^2(\mathbb{R}^{2n}\smallsetminus\{0\})$; $H$ is positively homogeneous of ...
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5answers
1k views

Why the surface of the sphere is not a Euclidean space?

In wiki, I see a description in manifold: "Although a manifold resembles Euclidean space near each point, globally it may not. For example, the surface of the sphere is not a Euclidean space, but in ...
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1answer
49 views

What does |AM|=|AC| mean?

What does $$|AM|=|AC|$$ mean? $$AM$$ and $$AC$$ are rays. Do they mean the length or what?
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2answers
79 views

Finding the ratio of diagonal segments of quadrilateral given four sides and a diagonal.

Here is a picture of the question: $ABCD$ is a quadrilateral. $[AC]\cap[BD]=E$. $|AB|=11$. $|BC|=16$. $|CD|=13$. $|AD|=12$. $|AC|=15$. What is $\frac{|DE|}{|BE|}$? This ...
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0answers
21 views

Is there a converse for the equal intercept theorem?

If equal intercepts are made by transversals on three or more lines, then are the lines parallel to each other?
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2answers
33 views

Is it possible to bound the hypotenuse given the perimeter of an orthogonal triangle?

I am working on Project Euler's problem 9, which needs you to calculate the product of a pythagorean triplet which sums to 1000. Therefore we have: $a < b < c$ $c^2=a^2+b^2$ $a+b+c=1000$ I ...
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1answer
19 views

Rate of change of area depending of the circumference of a circle

I want to find the general formula which gives the rate of change of the area with respect to the circumference. (Of a circle) I know that I can use differentiation formulas but I don't want to do it ...
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1answer
22 views

Calculate the distance from Plickford to Murbell

Attached is my question. Please provide an explanation for how I could calculate the distance from Plickford to Murbell.
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1answer
38 views

Find ratio of areas of triangle to pentagon?

ABCDE is a regular pentagon; rays AB and DC intersect at X. Now the area of triangle BCX is 1. What is the area of the pentagon? I figured out that the area of the pentagon is the square root of 5. ...
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1answer
30 views

Find the magnitude of the angle

It's a nice problem and I have got a little bit and have some troubles.Could you please help me in finding the answer for the following problem? Let $ABCD$ be a convex quadrilateral such that $\hat{ ...
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1answer
26 views

Number of Pieces a regular $n$-gon is cut into by its diagonals [closed]

In how many pieces a regular n-gon is cut into by its diagonals? I need a general formula. By inspection, I have the solution to some lower values of $n$. For $n=3,4,5,6$ solutions are $1, 4, 11, ...
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2answers
12k views

coordinate geometry: finding the ratio in which a line segment is divided by a line

The question is: Determine the ratio in which the line 3x + 4y - 9 = 0divides the line segment joining the points ...
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2answers
23 views

Proof that a secant line intersects a circle in exactly two points (according to Hilbert's axiomatic system)

With Hilbert's axiomatic system, How do I prove that a non-tangent line $d$ that intersects a circle $C$ intersects it in exactly two point? My teacher gave us the following clue: First show that if ...
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0answers
11 views

Is the angle between the most outer vertices bigger or smaller then $\pi$

I have multiple vertices with a common startpoint on a plane in $\Bbb R^3$, so the problem can be simplified to $\Bbb R^2$. I want to check if the angle between the two most outer vertices is bigger ...
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0answers
17 views

Finding rotation and translation of a planar object in 3d

I have a planar object, say a polygon $A_1A_2\ldots A_n$ in the 3-dimensional Euclidean space. It is translated by a vector $v$ and rotated by a rotation matrix $R$, and the resulting image is ...
2
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0answers
95 views

What is the best approximate of points on a sphere?

I have a unit radius sphere with a set $S$ of $n$ points on it. How can I find a map $f:S\to \mathbb{R}^4$ which minimizes $$\sum_{x,y\in S} \bigg( d_{\text{geodesic}} (x,y)^{2} - ...
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1answer
30 views

construction of line segment of a length $\sqrt{a^2-b^2+c^2+d^2}$

There are line segments $a, b, c, d$ and $a > b$. I have a question how to construct a line segment of a length $\sqrt{a^2-b^2+c^2+d^2}$. I can use Pythagoras theorem but I don't know how to make ...
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1answer
27 views

Tricky collinear vectors problem

Let OABC be any planar quadrilateral. Let $G_1, G_2$ and $G_3$ be the centroids of OAB, OBC and OAC respectively, and let G be the centroid of the triangle $G_1G_2G_3$. Show that the points O, G and ...
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2answers
46 views

Geometry problem involving orthocentre and midpoint of sides.

Let $AA_1, BB_1, CC_1$ be the altitudes of $\Delta ABC$ and let $AB \neq BC$. If $M$ is midpoint of $BC$, $H$ the orthocentre of $\Delta ABC$ and $D$ the intersection of $BC$ and $B_1C_1$, prove ...
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1answer
43 views

Is there a problem of plane geometry whose analytic reformulation gives a polynomial non-solvable by radicals?

This answer explains that any elementary plane geometry problem can be reduced to the existence of a solution of a polynomial system (called the analytic reformulation). Question: Is there a ...
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30 views

Geometry (using vectors or complex numbers) problem involving points on a triangle and on a circle

Let ATS be a fixed acute-angled triangle, i.e., all the three angles of the triangle are less than 90 degrees. Let E and F be two points on the sides AS and AT, respectively, such that the segment EF ...
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182 views

Assumptions needed for proof of the Pythagorean Theorem from examples

There are lots of geometric dissection and reassembling proofs of the Pythagorean Theorem (PT). I want to know what are the necessary ideas to allow a proof using discrete instances. For example, we ...
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0answers
24 views

How to embed points on a sphere in a 3 or 4 dimensional space

I am looking for a procedure to embed points on a sphere in a 3 or 4 dimensional Euclidean space such that the distances are preserved as much as possible. If there is any related optimization ...
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0answers
37 views

Apostol's Mathematical Analysis: On the Geometric Representation of Real Numbers

In Apostol's Mathematical Analysis (second edition), it is written, on p.3: The real numbers are often represented geometrically as points on a line (called the real line or the real axis). A ...
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1answer
51 views

Condition for three lines to be concurrent.

A triangle $\bigtriangleup ABC$ is given, and let the external angle bisector of the angle $\angle A$ intersect the lines perpendicular to $BC$ and passing through $B$ and $C$ at the points $D$ and ...
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0answers
49 views

Circle continuty principle proof

Circular continuity principle: If a circle C has one point inside and one point outside another circle C' , then the two circles intersect in two distinct points. I read this on Euclidean and ...
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1answer
48 views

how to measure a circle.

As we all know we can measure a line with a scale or any instrument but right now I have studied circles and was wondering if there was any way , instrument or method to measure circle i.e the ...
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1answer
23 views

Cyclic hexagon with every other side equal

Let $ABCDEF$ be a cyclic hexagon with $AB=CD=EF$. Let $AC\cap BD=P, CE\cap DF=Q, EA\cap FB=R$. Prove that $\triangle PQR\sim\triangle BDF$. This problem seems simple, but I'm having trouble figuring ...
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1answer
29 views

Find point $E$ on $CD$ of parallelogram $ABCD$ such that $\angle AEB = \angle BEC$

Find point $E$ on $CD$ of parallelogram $ABCD$ such that $\angle AEB = \angle BEC$ Shape is supposed to look something like this.
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1answer
28 views

Proof about uniqueness of point $P$ such that its power to two circles is equal.

I've tried to prove that there exists only one point $P$ on $O_1O_2$ such that $Pow(P,O_1)=Pow (P,O_2)$ where $O_1 $ and $O_2$ are circles with no point of tangency and I've got the following ...
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1answer
24 views

A problem on Euclidean Geometry

How can i prove that if a triangle has sides of lengths a, b, e, then its area S satisfies the inequality $$4\sqrt{3}\leq a^{2}+b^{2}+ c^{2}$$ with equality holding only for equilateral triangles. ...
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1answer
29 views

Hint on solving a problem in Euclidean geometry

How do i prove the following problem: If a quadrilateral has sides of length $a$, $b$, $c$, and $d$, prove that its area $S$ satisfies the following inequality $$4S\leq (a+c)(b+d)$$ with equality ...
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1answer
86 views

The Undefined Terms in Geometry (i.e. points, lines & planes)

As I understand it, there are three undefined terms (alternatively they are sometimes called primitive notions) in Geometry: Point: A point has 0 dimensions and merely denotes a location. Line: A ...
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1answer
30 views

An exersise of Euclidean geometry

The following question is one of the exercises "Foundation Euclidean and non-Euclidean geometry" by Greenberg (chapter 1/ Major Exercises/ 3 ) For any angle, draw a circle $\gamma$ radius $d$ ...
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2answers
90 views

Concurrency of lines formed by pair of circles joining pairwise.

How can i prove that lines $AD,EB,CF$ are concurrent ? My attempt Considering $\Delta ACB$ I've got the condition that $\cfrac {CP \cdot BQ \cdot AR}{PB \cdot AQ \cdot RC}=1 $ ,but I don't see how ...
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1answer
51 views

Find an angle in a triangle with cevians

Given triangle ABC such that angles B and C both measure 70 degrees, points E and F lie on sides AB and AC, respectively, such that angle ABF measures 30 degrees and angle ACE measures 50 degrees. ...
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0answers
12 views

Calculate Distance between a point and plane in 3 Space and determine the relationship

I am asked to find the distance between a point and plane given the following Point $(1,1,0)$ Plane $2x - 3y + 6x = -1$ Now I use the formula And ultimately by substituting in a arrive at ...