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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Find the maximum possible area for the triangle

Two vertices of an isosceles triangle are (1,2) and (4,6). The inradius of the triangle is $\frac{3}{2}$. Find the maximum possible area for the triangle. My work, for the two possible structures of ...
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1answer
37 views

slope of a curve in $\mathbb{R}^3$

The surface given by $z = x^2 -y^2$ is cut by the plane given by $y = 3x$, producing a curve in the plane. Find the slope of this curve at the point $(1, 3, -8)$. My answer is: $$f(x, y, z) = x^2 - ...
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1answer
30 views

Line goes through the center of circle [closed]

How can I prove this theorem which looks easy. Circle $C$ has a center $O$. Take four points $A,B,D,E$ on the circumference of $C$ such that $AB$ is parallel to $DE$. Let $F$ and $G$ be midpoints of ...
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1answer
28 views

Prove that $\angle BAC + \angle OAP = 180^\circ$

Prove that if you construct two circle centered at O and P and intersecting at A with tangent lines BA and CA. Prove that $\angle BAC + \angle OAP = 180^\circ$. I'm having trouble just starting the ...
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0answers
43 views

Derivation of Steepest Descent Direction used in Line Search Methods

In the numerical optimization text I am reading, the Steepest Descent Direction was derived by considering $$ \min_{||p||_2\leq 1} p^T\nabla f(x_k) $$ This resulted in $$ p_k=-\frac{\nabla ...
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0answers
35 views

Synthetic proof of curvature formula

The radius of curvature of a curve $\gamma:I \to \mathbf{R}$ parametric by arc length is $||\ddot \gamma||^{-1}$. I want to demonstrate this using synthetic geometry. Let $A$, $B$ and $C$ be three ...
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0answers
17 views

How To Find a Set of Points Farthest Apart Within 3D Solid

I am trying to find out a method to solve the following problem: There are two parameters: 1) There is a solid 3D region plotted in a cartesian coordinate system. 2) There is a number of points that ...
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4answers
299 views

How to show that these two lines are perpendicular?

Let $AEE'$ be an isoceles triangle with $\angle EAE'=90^\circ$ such that $AE=AE'$ and such that $A$, $E$ and $E'$ lie on the circle $c_1$. Let $ADD'$ be an isoceles triangle with $\angle ...
0
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1answer
51 views

Chords joining midpoints of four arcs on a circle

Let $W$, $X$, $Y$, $Z$ be the midpoints of arcs $\stackrel{\frown}{AB}$, $\stackrel{\frown}{BC}$, $\stackrel{\frown}{CD}$, $\stackrel{\frown}{DA}$, respectively. Show that the chords $WY$ and $XZ$ ...
3
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1answer
28 views

How to find these quantities so as to conform to these conditions?

Suppose $a \in \mathbb{R}^k$, $b \in \mathbb{R}^k$. Then how to find $c \in \mathbb{R}^k$ and $r > 0$ such that the following holds? For any $x \in \mathbb{R}^k$, we have $$|x-a| = 2 |x-b|$$ if ...
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1answer
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How to prove this assertion about $\mathbb{R}^k$?

Suppose $k \geq 3$, $x$, $y \in \mathbb{R}^k$, $|x-y| = d > 0$, and $r > 0$. Then how to prove the following assertions? (a) If $2r > d$, then there are infinitely many $z \in \mathbb{R}^k$ ...
2
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1answer
32 views

Find the equation of line and finding a point in given example

The outer circle is $x^2+y^2=1$ and the smaller circle is $x^2+(y+1-r)^2=r^2$. The arclength is parameterised anticlockwise with $s=0$ at the bottom as shown. If we know $s_n$ and $s_{n+1}$ can we ...
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1answer
45 views

Relationship between the altitude of an isosceles triangle and segments drawn to the lateral side from a point on the base.

Question :In an isosceles triangle, the sum of the distances from each point of the base to the lateral sides is constant. I've tried a couple of things, but it seems like this statement is not true. ...
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0answers
12 views

Name of the set of points equidistant from a line

I was reading about geometrical shapes in n-dimensional Euclidean spaces and programming some objects that would share some of their properties in different dimensions, like n-spheres. I had somewhere ...
4
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4answers
807 views

Why 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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2answers
36 views

Rational distance implies countable set

I am working on this problem for weeks without a good solution. Let $S\subset\Bbb R^d$ be a set in which $\rho(s_1,s_2)\in\Bbb Q$ for any $s_1,s_2\in S$, where $\rho$ is the Euclidean distance in ...
3
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1answer
31 views

Prove that $\max\{|x_i|: 1 \leq i \leq n\} \leq \|\vec{x}\| \leq \sum_{i=1}^{n} |x_i|$

If $\|\vec{x}\|$ denotes the Euclidean Norm of $\vec{x} \in R^n$, show that $$ \max\left\{|x_i|: 1 \leq i \leq n\right\} \leq \|\vec{x}\| \leq \sum_{i=1}^{n} |x_i| $$
0
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1answer
20 views

On inscribed quadrilateral

For $i=1,2$, let: $\Gamma_i$ two circles intersecting each other at $A,B$. $r$ a line containing $A$ intersecting $\Gamma_i$ at $T_i\neq A$. $t_i$ tangent line to $\Gamma_i$ at $T_i$. $P=t_1\cap ...
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1answer
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Prove that the vector sum of the vertices of an n-sided regular polytope whose center is at the orgin is zero

I need to prove this (assuming it's true): The vector sum of the vectors pointing to the vertices of an n-sided regular polytope whose center is at the origin of a Euclidean space is zero. If it ...
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0answers
56 views

When do two points coincide in euclidean geometry?

The 4° common notion in the Elements of Euclid says: "Things which coincide with one another equal one another". Many authors have interpreted this sentence as a principle of superposition that could ...
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38 views

prove that there exists a line separating all points in x by color.

Please help to solve this question..! Let $X\subset\mathbb{R}^2$ be a set of $n\geq4$ points in general position. Suppose points in $X$ are colored with two colors, such that for every four points ...
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Prove that there exist 4 such planes which form a tetrahedron containing P.

Let $P \subset \Bbb{R}^3$ be a convex polytope such that of the planes spanned by the faces, every $3$ intersect at a point, but no $4$ intersect. Prove that there exist $4$ such planes which form a ...
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1answer
57 views

prove that all line can be intersected with a unit circle

I need to solve this problem.. from geometry. suppose there are n lines in the plane such that every three of them can be intersected with a unit circle. Prove that all of them can be intersected ...
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2answers
301 views

Interesting locus

Consider an acute triangle $ABC$ and non-constant(*) point $P$ on $AB$. Take then points $D$ and $E$ on $AC$ and $BC$ respectively such that $\angle DPA=\angle EPB=\angle ACB$. Let $M$ be the ...
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find the side of this equilateral triangle with an point inside.please help me. [duplicate]

Triangle $ABC$ is an equilateral triangle and $P$ is an arbitrary point inside it. The distance from $P$ to $A$ is $4$ and the distance from $P$ to $B$ is $6$ and the distance from $P$ to $C$ is $5$. ...
0
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1answer
54 views

Determine all the integer solutions to $23x + 39y = 2$

I would like to calculate all the solutions to this equation using Euclides' algorithm and linear combination after finding the GCD. I suppose it's easy, but I'm a beginner. $23x + 39y = 2$
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1answer
36 views

Test if a point is inside a 3D cuboid

I have a cuboid in 3D space, it is not regular at all. I do have the coordinates of its 8 vertices and my problem is how to determine a given point coordinate is inside or outside this cuboid. I ...
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Prove Proposition I.7 of Euclids for the case where D lies in the interior of triangle ABC

Prove that Given triangle ABC with apex C, we cannot construct another triangle ABD with D lying in the interior of ABC Is this proven the same way as if D lies on the same side of AB? I have it ...
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2answers
54 views

Trisecting a Triangle

Given a (non-degenerate) triangle $PQR$ in the Euclidean plane, does there exists a point $A$ in the interior of the triangle such that, the triangles $APQ$, $AQR$, and $ARP$ have same area? If it ...
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Number of faces $F$, edges $E$ in a polyhedron.

A polyhedron has the property that three edges intersect in six of its nodes, while four edges intersect in the remaining node, what are the number of edges and faces in the polyhedron? I've gone ...
2
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3answers
83 views

Diameter of a circle touching three inner circles of diameter 1

If the diameters of three three inner circle are $1$ meter, what is the radius of the big circle? (Note: the OP provided own answer below, after getting a hint).
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1answer
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Bisectors problem

I need help with this geometric problem. Given triangle ABC. CM is the bisector of $\angle ACB, M\in AB$ and $CN, N\in AB$ is the bisector of the suplementary angle of $\angle ACB$. B is between M and ...
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1answer
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Calculating weighted Euclidean distance with given weights

This question is regarding the weighted Euclidean distance. I have three features and I am using it as three dimensions. I need to place 2 projects named A and B in this 3 dimensional space and ...
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0answers
78 views

Similar Triangles Proof - How to tackle proofs?

Well, I know it is repetitive.I have read the proof from different textbooks.But sometimes I feel doubtful about it all.Every time I try to prove it for myself, I fail at some points.I'm asking those ...
0
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1answer
41 views

Projecting line onto edge of ellipse?

I feel like the answer to this should be fairly simple, but I am absolutely hitting a brick wall here. I have a line, with angle $\beta$ and origin $(x_l, y_l)$. I have a rotated ellipse, with major ...
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2answers
89 views

The concurrence of angle bisector, median, and altitude in an acute triangle

$ABC$ is an acute triangle. The angle bisector $AD$, the median $BE$ and the altitude $CF$ are concurrent. Prove that angle $A$ is more than $45$ degrees. Here $D,E,F$ are points on $BC,CA,AB$ ...
2
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1answer
37 views

Difference between Euclidean space and $\mathbb R^3$

What is the difference between Euclidean space and $\mathbb R^3$? I have found in some books that they are the same, but in other references like Wikipedia, it says that a vector in $\mathbb R^3$ is ...
4
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1answer
44 views

Sum of Determinants = Scalar product with normal vector?

Today I have seemingly simple question and maybe someone knows the answer without getting into messy calculations. So we have $n$ vectors $v_1,\dots,v_n\in\mathbb{R}^n$ and let us assume for the ...
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4answers
70 views

Two overlapping squares

$ABCD$ is a square. $BEFG$ is another square drawn with the common vertex $B$ such that $E,\ F$ fall inside the square $ABCD$. Then prove that $DF^2=2\cdot AE^2$.
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1answer
66 views

Shortest path between two points via two disks

Hallo everybody, I have the following problem regarding shortest paths in $R^2$. Suppose you are given two points $p$ and $q$ and two unit disks, as in the picture. I am looking for a path from ...
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1answer
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exercises for Euclid's Elements

Can you suggest some books with exercises related to Euclid's Elements, or to Euclidean Geometry, as an aid to an undergraduate course on Euclidean Geometry and its history? I need exercises that ...
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1answer
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Geometry question pertaining to a plane going through the skeleton of a cube

My question is as follows: a plane that has taken the shape of a pentagon is intersecting the skeleton of the cube. Or I guess we could think of it as a cross section. Points $M$ and $N$ were used ...
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Which statements are equivalent to the parallel postulate?

I would like to have a long-ish list of statements that are equivalent to the parallel postulate. If a line segment intersects two straight lines forming two interior angles on the same side that ...
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There exist a bijection between the Real numbers and points of a straight

Assuming that we are building our geometry on the axioms of Euclid/Hilbert, and using either the Dedeking or Cauchy construction of the reals, how can one prove this statement? I've looked up on the ...
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2answers
86 views

Visually understanding the formula for the distance from a point to plane.

Ok, so we know that if we have an arbitrary point, $p$, and a normal perpendicular to an arbitrary plane, $n$, the distance from the point to the plane can be computed as follows: $$distance = p ...
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1answer
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Is there a name for this point?

I found the following problem interesting: In a three villages $A$, $B$ and $C$ there are $a,b$ and $c$ pupils respectively. Where should one build the school such that the total length of pupils ...
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0answers
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Propagating a 3d vector to a spcific point in a 2d plane

I have an $xyz$ point $P$, and a 3d vector pointing from it denoted by $N$. I want to propagate the vector forward to a certain point in the $xy$ plane and calculate the corresponding value of $z$. ...
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2answers
52 views

Find the value of $a$.

please help I'm lost on what numbers to add or what formula to use
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1answer
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I need to find the value of x. Im only given the a degree how would you solve this?

this is the link to the triangle that is connected to the question. What is the value x?
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2answers
51 views

How to find the area of an isosceles triangle without using trigonometry?

I have an isosceles triangle with equal sides $10$ unit, angle between them is $30^\circ$. I need to be confirmed that the area of this triangle can be found in any method other than using any kind ...