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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A straightedge and compass construction: $\left(\widehat{A},r,b-c\right)$

I am looking for an elegant solution of the following problem: Construct $ABC$ with straightedge and compass, given $\widehat{A},r,b-c$. By taking the lines $AB,AC$ as a skew reference system, ...
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18 views

Vector of triangle height constructed over two vectors

Given vectors: $\overrightarrow{a}=\overrightarrow{p}+2\overrightarrow{q},\overrightarrow{b}=3\overrightarrow{p}-\overrightarrow{q}$ where ...
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36 views

Suppose that $ABCD$ is a trapezoid with $AB$ parallel to $CD$.

Suppose that $ABCD$ is a trapezoid with $AB$ parallel to $CD$. Let $P$ be the point where the diagonals $AC$ and $BD$ intersect. Show that the triangles $CDP$ and $ABP$ are similar. Use this to prove ...
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24 views

How do I make an inner rounded rectangle and an outer rounded rectangle be parallel around the corners?

The outer radius does not follow the inner radius. I am currently using x = width/4 + radius + outset, y = height/4 + radius + outset. I think the outset needs to be some ratio of the ...
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1answer
28 views

Suppose that $a, b,$ and $c$ are distinct points in $\mathbb{C}$.

Suppose that $a, b,$ and $c$ are distinct points in $\mathbb{C}$. Let $l$ be the line which bisects the angle $\measuredangle bac$. For a point $z$ on $l$, let $p$ be the point on the line through $a$ ...
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29 views

Suppose that $a, b$, and $c$ are distinct points in $C$. Suppose that $l$ is the line which bisects $∠bac$ and $m$ is the line which bisects $∠acb$

Suppose that $a, b$, and $c$ are distinct points in $C$. Suppose that $l$ is the line which bisects $∠bac$ and $m$ is the line which bisects $∠acb$. Now let $z$ be the point $l \cap m$. Let $n$ be the ...
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134 views

Moving Line Segment Problem

So I was contemplating the following problem: Given two circles of radius $r$, which will position vertically so that their centers are separated by a distance of $2r+d$ (and thus, at the nearest ...
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1answer
62 views

condition for cones to be reciprocal

Question : Show that the cone $$ax^2 + by^2 + cz^2 - cxy - ayz - bzx = 0$$ is the reciprocal of the cone $$(a^2 - bc)x^2 + (b^2 - ac)y^2 + (c^2 - ab)z^2 - 2(a^2 + bc)yz - 2(b^2 + ac)zx - 2(c^2 + ab)xy ...
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20 views

Find $a\in\mathbb{C}$ so that the equation of the line through $−2 + i$ and $−2i$ is $z\bar{a} − a\bar{z} = 8i$

Consider the following geometry problem: Find $a\in\mathbb{C}$ so that the equation of the line through $−2 + i$ and $−2i$ is $$z\bar{a} − a\bar{z} = 8i$$ where the bar represents the complex ...
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27 views

Distance between two Polar-Coordinates

I choose two Points in Berlin with the coordinates: 1: lat: 52.511206 long: 13.546486 2: lat: 52.527501 long: 13.319206 With an online tool I got the ...
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4answers
67 views

Construct a triangle given certain lengths related to a bisector

Let $ABC$ be a triangle, and $AD$ the bisector of angle $A$. Write $AB = c$, $AC = b$, $AD = d$, $BD = c'$, $CD = b'$. Using ruler and compass, construct the triangle $ABC$ given the lengths $d$, ...
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1answer
55 views

Origami: What shapes are impossible?

Let's say we had a collection of pieces of rectangular paper of any size to choose from. Using one sheet only of any chosen size, what three-dimensional (or two-dimensional) shapes are impossible to ...
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1answer
56 views

Does this line preserving, continuous bijection on an equilateral triangle exist?

I'm trying to define a continuous bijection on the points (x/y coordinates) of an equilateral triangle. The vertices and the midpoints of each edge need to remain fixed. The (6) points on the edges ...
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13 views

Sphere overlap with cubic grid in R^N; minimal number of grid cells

I have: n-dimensional sphere of radius $r<0.5$ with position $\vec x $ generated randomly from uniform distribution $M$ different $N$-dimensional cubic grids, all with cell size $L=1.0$ which has ...
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1answer
40 views

Identifying the centre of a circle

I've been asked to identify with reasons of properties of circles , the centre of the circle which passes through Q1 . A, D and E Q2. A, C, And E . I'm not too sure how to approach this question ...
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2answers
42 views

How to find whether a point lies on a line which is in parametric form?

Does the point $(1,8,3)$ line on the line with parametric equation: $$x = 5 + 2t$$ $$y = 2 + 6t$$ $$z = 1 + 3t$$ I know how to solve if they give me a equation of a plane and ask whether ...
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55 views

Properties of circles

I'm told to find angle CAD and I got stuck... Given that CD = BC My workings .. CAT = ACT = $180-40/2 = 70$ CAP = $180-70-60=50$
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1answer
22 views

Finite points in geometry

What are finite points in Geometry? I was reading this and they mention "Suppose that $P$ and $U$ are finite points having normalized barycentric coordinates $(p,q,r)$ and $(u,v,w)$." I am ...
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0answers
78 views

A chain of six circles associated with a conic

I am looking for a solution of the following problem: Let $A_1, ..., A_6$ and $B_1, ..., B_6$ be 12 points lying on a conic, and suppose that for $i=1, ..., 5$ through $A_i, A_{i+1}, B_{i+1}, B_i$ ...
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3answers
139 views

Construct a parallelogram subject to certain conditions

I am having trouble with the following exercise from Dollon and Gilet's Géométrie plane. Two parallel lines $\Delta$ and $\Delta'$ are given, as well as a point $A$ on $\Delta$ and a point $O$ on ...
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1answer
40 views

Finding angle associated with point inside an equilateral triangle.

$\triangle{ABC}$ is an equilateral triangle. $|AD|=6$. $|BD|=10$. $|CD|=8$. What is $m\angle{CDA}$? First thing comes to mind is Ceva theorem. I used its trigonometric form to reach ...
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1answer
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Intersection of line and plane [closed]

I'm trying to figure out a problem in perspective geometry. Given the line from the origin to the point $P(x,y,z)$ at some point in space it casts a picture of a point when intersecting the plane ...
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1answer
25 views

Area of quadrilateral with equilateral triangles erected on sides (

Let $ABC$ be a triangle. Construct points $B'$ and $C'$ such that $ACB'$ and $ABC'$ are equilateral triangles that have no overlap with $\triangle ABC$. Let $BB'$ and $CC'$ intersect at $X$. If $AX ...
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4answers
58 views

Definition of an angle

I saw two definitions of an angle. Are those equivalent or is another wrong in some axiomatic system? An angle is the union of two rays. An angle is a subset of a plane restricted by two rays. I ...
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20 views

How to construct a polyhedron from given planes

This seems to be a basic questions, but I really don't know a good computer algorithm to do this. I have a set of planes (parameterized by normal direction and distance from a given point), and I want ...
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1answer
30 views

4 known points, 8 unknown - problem of uniqueness of solution

I have 4 known points $\textbf{P}_i \quad (i = 1, 2, 3, 4)$. One of these points is simply $\textbf0$. I have 8 unknown points: $\textbf{F}_j$ and $\textbf{R}_{jk} \quad (j = 1,2; \quad k = 0,1,2)$. ...
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1answer
33 views

What kind of line is formed when a piece of paper is folded?

What kind of line is formed when a piece of paper is folded? I am 17, and I'm homeschooled. I have a book from 1958 that's called "Plane Geometry, Welchons Krickenberger Pearson" It has no answers in ...
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2answers
206 views

Geometrical proof required regarding midpoints

$ABCD$ is a convex quadrilateral with $W, X, Y, Z, M$ and $N$ as the midpoints of $AB, BC, CD, DA,$ the diagonals $AC$ and $BD$ respectively. [Then, $WXYZ$ is a parallelogram with $K$ as the ...
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How to prove that this is a parellelogram?

Prove that ABED is a parallelogram Given: ABCD is a trapezium F and G are the midpoints of AB and DC respectively FHG is a straight line AD is equal to and parallel to BE My attempts have ...
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1answer
56 views

Trisecting an an angle - how to prove?

I have this question on my "History of Mathematics" problem set: Draw any angle AÔB; Pick a point C on OB; Now trace CD which is perpendicular to OA; Draw a parallel line $s$ to OA which passes ...
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2answers
66 views

Three Altitudes of a triangle are concurrent

I have been told that this well known fact can be shown using only Euclid's propositions from books one to three, and cyclic quadrilaterals. I can't figure out how to start, which quadrilateral ...
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2answers
45 views

Distance inequality

Consider $H=\{ (x,y) \in {\bf R}^2\mid x$ or $y$ is an integer $\}$ If $d$ is canonical distance in ${\bf R}^2$, show that if $d(x):=d(x,H)$, (1) $$ d(x) - d(y) \leq d(x,y) $$ if $x,\ y$ are in same ...
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1answer
15 views

Show that if $Q'$ is any point on the line of action of $F$, then $PQ × F$ = $PQ'× F$

If a force $F$ is applied to an object at a point $Q$, then the line through $Q$ parallel to $F$ is called the line of action of the force. We defined the vector moment of $F$ about a point $P$ to be ...
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28 views

Area enclosed by a polygon

I did some work in the area of mensuration and came across an interesting concept/formula. The formula states that For a polygon having vertices $(x_1,y_1),(x_2,y_2),(x_3,y_3),\dots(x_n,y_n)$, the ...
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18 views

Converse of a generalization of the Simson line

When I construct theorem 1 in, I found a converse of this theorem as follows, I am looking for a proof. Let $ABC$ be a triangle, let $P$ be a point on the circumcircle of $ABC$. Let $H$ is the ...
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1answer
66 views

Clever way to sum these angles? [closed]

In the image, is there a nice way to write down the sum of a+b+c?
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1answer
46 views

A problem to show a certain sum is invariant in some Euclidean geometric configurations.

$\textbf{Problem.}$ Suppose circles of radius $r$ and radius $s$ are externally tangent at the point $1/2$ and internally tangent to the unit circle. There are infinitely many such configurations, one ...
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27 views

Questions about Scale Factor?

We have a final study guide for our geometry class and we never learned anything about scale factor so we were just wondering if anyone could help explain any of the questions. One of the people in ...
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42 views

New circles associated with the Miquel theorem configuration

Consider the Miquel six circle theorem configuration Let $A, B, C, D$ are concyclic, $A_1, B_1, C_1, D_1$ are concyclic, and $A, A_1, B_1, ,B$; $B, B_1, C_1 C$, $C, C_1, D_1, D$; $D, D_1, A_1, A$ ...
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2answers
45 views

Arbitrary Dot and Cross Products

I am having a bit of trouble with answering these few dot and cross product questions. Suppose that $u · (v × w) =3$. Find, $w · (u × v)$ $v · (u × w)$ $(u × w) · v$ Could some explain their ...
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1answer
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When an arbitrary point belongs to a given pyramid? [duplicate]

Suppose that we fix points $a,b,c,d \in \mathbb{R}^3$ which do not belong to the same plane. How can we check algebraically whether $x \in \mathbb{R}^3$ to their convex hull?
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26 views

Orthographic projection - calculate the matrix based on lengths

To simply describe the situation: You have a photograph (taken from unknown angle or distance) of a flat pitch on which two straight sticks of known length were randomly placed. The goal is to ...
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1answer
23 views

Showing the distance between a point $P$ the line determined by a segment $AB$ is $d=\frac{||AP\times AB||}{||AB||}$

Show that in $3$-space the distance $d$ from a point $P$ to the line $L$ through points A and B can be expressed as $$d=\frac{||AP\times AB||}{||AB||} .$$ My diagram of the situation: My next ...
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2answers
133 views

Construct 60° angle through point, other line in only four compass-and-straightedge steps

PROBLEM Here is a surprisingly intriguing challenge posed on Euclidea, a mobile app for Euclidean constructions: Construct a 60° angle through both a point $P$ and an external (infinite) line ...
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1answer
45 views

Golden ratio in regular pentagon

Let $ABCDE$ be a regular pentagon, F in arc small BA. Show that $\frac{FD}{FE+FC}=\frac{FB+FA}{FD}=0.618033$ the golden ratio; and $FD+FB+FA=FE+FC$
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1answer
32 views

Given a diameter of a circle bisecting the angle formed by two intersecting chords, Prove the chords are equal

In a circle, a diameter bisects the angle formed by two intersecting chords. Prove that the chords are equal
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35 views

Show that any quadrilateral can become cyclic by changing angles.

Given a convex quadrilateral with given side lengths $a$, $b$, $c$, $d$ in that order (meaning that side a intersects $d$, $b$; side $b$ intersect $a$, $c$ e.t.c.), show that it is possible to "make" ...
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19 views

Equation with L2-distance of two single values

I am studying the equation which includes the L2-distance term $||a_n - b_n||_2$ (taken from http://caffe.berkeleyvision.org/doxygen/classcaffe_1_1ContrastiveLossLayer.html). Here, $a$ and $b$ are ...
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0answers
94 views

Problem on similar triangles in a weakened axiom system

In the figure above, $A'C'$ is parallel to $AC$. It is obvious, using similar triangles, that if $B$ is the midpoint of $AC$, then $B'$ is the midpoint of $A'C'$. I would like to know how easily ...
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0answers
15 views

Supporting hyperplane of a polarity of a convex body.

Recently, I am studying in combinatorial convexity and related topics. I use the book "Combinatorial Convexity and Algebraic Geometry" (GTM 168) as my main reference. The book is very good, all the ...