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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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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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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3answers
122 views

Is it possible to project orthogonally an ellipse with major and minor axes $2a$,$2b$ so that its image is a circle with diameter $2b$?

Problem: Prove that the area of an ellipse with major axis and minor axis of lengths $2a$ and $2b$,respectively, is $ab \pi$ . Proof: We do this by projecting the ellipse into a figure whose ...
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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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1answer
24 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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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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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
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
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
87 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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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 ...
3
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1answer
52 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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1answer
29 views

Maximize volume of hyperrectangle given fixed sum of components

Is there a named theorem that says for a k-dimensional hyperrectangle and a fixed sum $S$ of the side lengths in each dimension (what I called "components" in the title), the volume is maximal when ...
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0answers
32 views

Is equal curved surface areas a coincidence

If we take a cylinder of height $2x$ and radius $x$, as well as a sphere of radius $x$, we notice that they have the same curved surface area. Also, if we take the frustum of a cone such that it has ...
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0answers
22 views

Pointing at a specific coordinate on a plane

I am currently working on a school project that requires me to create a device that points a laser at any arbitrary point on a flat surface below it, by rotating it along the $X$ and $Y$ axis. The ...
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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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0answers
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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1answer
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Are lines which pass respectively through vertices $A,B,C$ and incenter, circumcenter and orthocenter of $\Delta ABC$ concurrent?

Prove that the lines through $A$ and the incenter of $\Delta ABC $, through $B$ and the circumcenter of $\Delta ABC$, and through $C$ and the orthocenter of $\Delta ABC $ are concurrent if and only if ...
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1answer
37 views

How to prove the concurrency of these lines?

Suppose that in $\Delta ABC$ we take a point $D$ on $BC$ such that the incircles of $\Delta ACD$ and $\Delta ABD$ are tangent at a point.Now suppose that points $H$ and $I$ are defined on segments ...
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8 views

Equations of an Oblique Circular Cone ($2$ circles are known)

I am trying to determine the parametric equations for an oblique circular cone with no success, as is shown in the figure above. Top circle is at point $(31,30,125)$ with a radius of $20$, and ...
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4answers
88 views

Which of the following lines are perpendicular to the line $3x + 2y = 7$?

The following are a list of 6 equations of lines. I am trying to determine which are perpendicular to the line $3x + 2y = 7$. This is not homework but a practice exercise for the GRE. (a) $y = ...
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1answer
25 views

Does a curve violate the first postulate of Euclidean Geometry?

Do the definitions and properties of curve secant and tangent lead to the violation of the first postulate of Euclidean Geometry?
2
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1answer
65 views

Constructing a square whose sides contain 4 given points

I'm interested in the following problem, from Terence Tao's "Solving Mathematical Problems": Exercise 4.3. (*) We are given four points A, B, C, and D. If possible, find a square so that each ...
4
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2answers
61 views

How to prove that these lines are concurrent?

Point $D$ is chosen on side $BC$ of $\Delta ABC$ such that the incircles of $\Delta ACD$ and $\Delta ABD $ are tangent at $G$. Let line $l$ be the angle bisector of $\angle ABC$ ,line $m$ be the ...
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2answers
59 views

Given locus is a circle, prove two lines are perpendicular

Let $l_1$ and $l_2$ be two lines in the plane. The locus of all points $P$, such that the sum of squares of the distances of $P$ to $l_1$ and $l_2$ is constant, is a circle. Prove that $l_1$ and $l_2$ ...
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2answers
32 views

Geometry with right triangles

I have this How do I find $h$? I know that I must use the cosine of $70$ degrees but I'm not sure how.
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28 views

How to determine that the area of a parallelogram in Euclidean Geometry

I can't seem to get that the area of a parallelogram is: $$Area = bh$$ I drew an arbitrary parallelogram ABCD and connected B to D. Then since I know the opposite sides of a parallelogram are ...
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1answer
43 views

Polyhedron cut along an edge

By cutting along an edge of a net of a polyhedron, you will form 2 pieces. Are there two distinct polyhedra for which this process may result in the same two pairs of pieces? Is there a real example? ...
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1answer
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How to prove Playfair's axiom or Euclid's parallel postulate without using angle measure

I have only seen this proof done using angle measure. How do you prove this without using angle measure or the fact that right angles are congruent?
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1answer
22 views

Max intersecting circle collection

Given any collection of $n$ distinct circles in the plane (with fixed but possibly distinct radii), is it always possible to rearrange the circles so that any two of them intersect twice?
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1answer
69 views

Proof of converse of Menelaus's Theorem

I am not understanding completely the last affermation of the following proof of the converse of Menelaus's Theorem. In the detail, the author, after having proven in general Menelaus's Theorem for ...
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1answer
49 views

How do I draw 3 colinear points only using a compass?

I know this is indeed possible because of the Mohr–Mascheroni theorem, and I thought about something like drawing two circles and looking at their intersection (for the 3rd point) but I have no idea ...
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0answers
37 views

Prove intersection between side length and tangent to circumcircle at opposite vertex is collinear with points on perpendicular bisectors of sides

Let $ABC$ be a triangle with $AB\neq BC$. Point $E$ lies on the perpendicular bisector of $AB$ such that $BE\perp BC$. Point $F$ lies on the perpendicular bisector of $AC$ such that $CF\perp BC$. Let ...
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2answers
28 views

Proving the possibility of rectangles within another

I was given this problem to prove yet Im having a hard time understanding what this problem actually is saying. Here is what I am understanding: In the plane we are given an infinite set of ...
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0answers
47 views

More problem on van Aubel configuration associated with parallelogram and cyclic quadrilateral

Problem 1: Let $ABCD$ be a parallelogram. Construct four squares on the sidelines $ABCD$. Let $NPMO$ be the Thebault’s square. Show that: 1-Centers of four circles $(NAP)$, $(PBM)$, $(MCO)$, $(OCN)$ ...
3
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3answers
61 views

What does $\mathbb{R}^\mathbb{N}$ actually mean?

We all know what $\mathbb{R}^n$ means. But I came across this statement about $\mathbb{R}^\mathbb{N}$ in a note that says $\mathbb{R}^\mathbb{N}$ is a vector space under pointwise operations has no ...
2
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1answer
23 views

What is the area of polygon DHEIFG?

In any triangle ABC, D, E and F are the midpoints of AB, BC and AC and from which perpendiculars are dropped on sides AB, BC and AC.The area of triangle ABC is S. Find the area of the polygon DHEIFG ...
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2answers
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How to prove three points are collinear when constructing a rectangle

My problem is: Choose a unit segment OI. Then construct a rectangle with base 3 units and height 2 units. I cannot use angle measure. I know I can construct this figure from my unit segment by using ...
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1answer
28 views

Problem involving the product of a secant and its exterior section that is created through a perpendicular to an extended diameter.

The question: The diameter AB of a circle is extended past B, and at a point C on this extension CD perpendicular to AB is erected. If an arbitrary point M of this perpendicular is connected with A, ...
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1answer
43 views

Concentric circles, and distances from one point to the endpoints of a diameter of the other.

If two circles are concentric, then the sum of the squares of the distances from any point of one of them to the endpoints of any diameter of the other, is a fixed quantity. I'm having a really hard ...
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1answer
37 views

Show that any vectors in $\Bbb R^2$ can be written as a linear combination of two orthogonal vectors.

I must show that any vectors $\vec{w}$ in $\Bbb R^2$ can be written as a linear combination of two non-zero orthogonal vectors $\vec{u}, \vec{v} \in \Bbb R^2$ using basic geometry concepts such as the ...
2
votes
2answers
61 views

Construction of a square ABCD

There are two nonparallel lines $p,q$ and point $A$, $A \notin p,q $ which lies between lines $p,q$. Construct a square ABCD such that $B \in p$ and $D \in q$. In special case in which $45°$ is angle ...
3
votes
1answer
25 views

Proving angles in the same corner equal

Suppose we have two line segments, AB and CD, which cross at point X. Now suppose there is an arbitrary point Y somewhere on the segment AX (that is, points A, Y and X are collinear). What is the ...
2
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1answer
130 views

Prove that the Cartesian product of two topological manifolds is a topological manifold.

I need help on the following problem, any responses would be greatly appreciated: Let $M$ be a topological $m$-manifold and $N$ be a topological $n$-manifold. Prove that $M \times N$ is a topological ...