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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3
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1answer
19 views

Question about concyclic points on the coordinate axes

If the points where the lines $3x-2y-12=0$ and $x+ky+3=0$ intersect both the coordinate axes are concyclic,then the number of possible real values of k is (A)1 ...
4
votes
2answers
37 views

External bisectors of the angles of ABC triangle form a triangle $A_1B_1C_1$ and so on

If the external bisectors of the angles of the triangle ABC form a triangle $A_1B_1C_1$,if the external bisectors of the angles of the triangle $A_1B_1C_1$ form a triangle $A_2B_2C_2$,and so on,show ...
6
votes
1answer
105 views

A movement requires 1 dimension, a rotation requires 2 dimensions, a what requires three dimensions?

Movement A zero-dimensional object cannot move. A one-dimensional object can move in one dimension (the x-axis). A two-dimensional object can move in two dimensions (the x-axis and y-axis). A three ...
2
votes
2answers
45 views

Prove line connecting intersection of tangents and opposite vertex bisects segment containing intersection of tangents and a vertex

Let $\triangle ABC$ be an isosceles triangle with $AB=BC$. Let $\Gamma$ be the circumcircle of $\triangle ABC$. Let the tangents at $A$ and $B$ intersect at $D$, and let $DC\cap\Gamma=E\neq C$. Prove ...
2
votes
1answer
45 views

One special case of Helly's theorem (for $\text{radius}=1$ circles)

There are $n$ points on the plane. Any $3$ of them can be covered with a radius $1$ circle. Prove that there is a radius $1$ circle that covers all the points. Came to this when tried to prove an easy ...
2
votes
4answers
87 views

Find the area of a triangle given the radius of its incircle and a tangential point

A friend gave me recently the following interesting problem and I would like to share a couple of solutions. Any additional contributions are welcome. A triangle $\vartriangle ABC$ is given and we ...
21
votes
3answers
1k views

A circle in the plane contains at most four lattice points?

Let $\cal C$ be a circle in ${\mathbb R}^2$ : $\cal C=\lbrace (x,y)\in{\mathbb R}^2 | (x-x_0)^2+(y-y_0)^2=r^2\rbrace$ for some constants $x_0,y_0,r$. What is the maximal number of points that can ...
3
votes
4answers
77 views

How to prove that the line perpendicular to the radius is the tangent in the calculus sense?

Let $P=(p_1,p_2)$ be a point on an semicircle and $r$ be the line perpendicular to the radius $\overline{OP}$, like the picture below. Euclid showed (Book III, Proposition 16) that $r$ does not ...
2
votes
2answers
31 views

Using the cross product to find the angle between two vectors in $\Bbb R^3$

Let $$u = \langle 1, −2, 3 \rangle \qquad \text{and} \qquad v = \langle −4, 5, 6 \rangle$$. Find the angle between $u$ and $v$, first by using the dot product and then using the cross product. ...
0
votes
0answers
29 views

Plotting Distance Constrained Points on a Plane

Does anybody know of some algorithmic way to tell if it is possible to plot a set of distance constrained points on a cartesian plane. Or, better still, a method to determine the minimum number of ...
2
votes
4answers
799 views

Two circles inside a semi-circle

Two circles of radius 8 are placed inside a semi-circle of radius 25.The two circles are each tangent to the diameter and to the semi-circle.If the distance between the centers of the two circles is ...
0
votes
1answer
40 views

equally spaced on circle question

Define $$\|\vec{x}\|:=\sqrt{\alpha^2+\beta^2},$$ where $\vec{x}:=(\alpha,\beta)\in \mathbb{R}^2.$ Set $$\mathbb{S}^1:=\{\vec{x}\in \mathbb{R}^2: \|\vec{x}\|=1\}\quad \quad and\quad \quad ...
4
votes
1answer
55 views

8 cubes ($2$x$2$x$2$) crossed by a straight line

There are 8 cubes forming a bigger cube whose dimension is $2$ x $2$ x $2$. Let a straight line (or a laser) try to pierce through as many small cubes as possible. At most how many small cubes can be ...
0
votes
1answer
21 views

Equivalence of euclidean and analytic geometry [closed]

I read about the axioms of euclidean geometry. How is analytic geometry rigorously defined? What are the axioms? And most important: How to prove that all the results proved in analytic geometry are ...
0
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0answers
17 views

Replacing Euclid's fifth postulate by any of two equivalent ones.

It is well-known that Euclid's fifth postulate has (at least) two equivalent substitutes. However, I fail to see how equivalent they are. How does one prove that the following postulates are ...
2
votes
1answer
15 views

Incident vector for lines in a 2D-Euclidean Geometry over Finite field

Consider the 2-D $EG(2,2^2)$ geometry. Let $\alpha$ be a primitive element of $GF(2^{2\times 2})$. The incident vector for the line $\mathcal{L} = \{\alpha^7, \alpha^8, \alpha^{10}, \alpha^{14}\}$ is ...
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4answers
51 views

Find the distance between the point $(0,0,0)$ and the plane $2x+3y+z=1$ [closed]

Find the distance between the point $(0,0,0)$ and the plane $2x+3y+z=1$. So I know in order to find the distance I need two points. How do I find a point in the plane?
2
votes
2answers
67 views

Triangle Geometry and Circles Problem

I have discovered something using Geogebra and I am positive it is true. I have tried to prove and my solution works but it is extremley convoluted. I'm hoping someone can provide a simple proof of ...
2
votes
1answer
62 views

Point on the Plane, a Triangle, and a Lower Bound of a Ratio Sum

Let $ABC$ be a triangle on the Euclidean plane. At which point $P$ on the plane does the ratio sum $\frac{PA}{BC}+\frac{PB}{CA}+\frac{PC}{AB}$ attain its minimum value? Prove also that, for any ...
7
votes
1answer
176 views

Why does the Pythagorean Theorem have its simple form only in Euclidean geometry?

Below are the right-angled forms of the Pythagorean Theorem in elliptic, Euclidean, and hyperbolic geometry, respectively. $$\cos\left(\frac{c}{R}\right) = ...
15
votes
1answer
234 views
+100

Hypercube and Hyperspheres

Let $n,k\in\mathbb{N}$. In this problem, the geometry of $\mathbb{R}^n$ is the usual Euclidean geometry. The lattice hypercube $ Q(n,k)$ is defined to be the set $ \{1,2,...,k\}^n ...
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0answers
19 views

what is exactly the fine-scale geometry of Euclidean space?

I have learnt math for a while, but I really do not have impressions on the concept of fine-scale geometry of Euclidean space. I hope this is not a trivial question as it sounds like. Any comments ...
1
vote
1answer
32 views

Distance between two 3D lines

What is the distance between the 3D lines $x = \begin{pmatrix} 1 \\ 2 \\ -4 \end{pmatrix} + \begin{pmatrix} -1 \\ 3 \\ -1 \end{pmatrix} t$ and $y = \begin{pmatrix} 0 \\ 3 \\ 5 \end{pmatrix} + ...
-1
votes
1answer
43 views

Point inside a triangle

let P be any point inside a triangle ABC.If $AP,BP,CP$ meet the sides $BC,CA,AB$ at points $D,E,F$ respectively,then prove that $PD+PE+PF< max(a,b,c)$ where $a,b,c$ are the lengths of the sides ...
10
votes
1answer
74 views

A generalisation of Napoleon's theorem. Is this result original?

I've found a generalisation of Napoleon's theorem to general polygons. Take any regular $n$-gon inscribed in a circle and stretch it (in any direction) so that the circle becomes an ellipse and the ...
0
votes
0answers
24 views

Characteristic polynomials for matrix A, involving the Identity matrix

Let us say we have a square matrix A, where A's characteristic polynomial is defined as $P_A(t) = \det (t I - A)$ (In this problem, I represents the identity matrix which has the same dimensions as ...
1
vote
1answer
98 views

Find A such that $A^2 \neq I$ but $A^4 = I$ [duplicate]

Find a $3 \times 3$ matrix A such that $A^2 \neq I$ but $A^4 = I$, where $I$ is the $3 \times 3$ identity matrix. Is there a simpler way to solve this problem rather than bashing it out by ...
1
vote
0answers
21 views

Is a composition of two $n-1$-dimensional symmetries a composition of $n-2$-dimensional symmetries?

Let $X$ be a finite dimensional real Euclidean space and $S,T$ be symmetries with respect to $n-1$-dimensional subspaces of $X$. Is it possible to write $ST$ as a composition of symmetries with ...
0
votes
4answers
38 views

Geometry question about lines

If I have two points in euclidean space or the Cartesian plane whichever and both points lie on the same side of a straight line. Both above or both below- how can I show that the segment connecting ...
0
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0answers
27 views

Is my proof of 'inscribed angle theorem' different from the usual one?

The usual proof of inscribed circle theorem uses the fact that the sum of interior angles is equal to exterior angle. I've formulated a little different approach, although the underlying concept is ...
2
votes
3answers
60 views

How to find the sum of distances so that it is minimal?

Question: $A$ and $B$ are two points on the same side of a line $l$. Denote the orthogonal projections of $A$ and $B$ onto $l$ by $A^\prime$ and $B^\prime$. Suppose that the following distance are ...
1
vote
1answer
18 views

Generalization of the Saccheri-Legendre Theorem Proof

So I'm working on generalizing the Saccheri-Legendre Theorem to convex $n$-gons. $\underline{\text{Saccheri-Legendre Theorem:}}$ The sum of the angles of a triangle is at most $180^\circ$. A ...
0
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0answers
18 views

When to use which condition number? (which norm)?

The condition number is used to determine how sensitive b is to changes in A in the equation ...
1
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0answers
37 views

What can be said about triangle with certain condition?

This question comes from 1988 Irish Mathematical Olympiad, for all those interested. A mathematical moron is given the values $b,c,\alpha$ for a triangle $ABC$ and is required to find $a$. He does ...
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3answers
36 views

Easy Compass Construction Problem

Here is a tricky compass and straightedge construction problem. Given triangle $\triangle ABC$ and point $D$ on segment $\overline{AB}$, construct point $P$ on line $\overleftrightarrow{CD}$ such ...
0
votes
1answer
25 views

Two circles covering the sides of a triangle

I would like to prove or find a counterexample for the following theorem: For any $\triangle ABC, \odot P_1, \odot P_2$ such that the three lines $AB, AC, BC$ are each contained in the union of the ...
0
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0answers
42 views

Countably Infinitely Many Points in a Euclidean Space

Do there exist $d\in\mathbb{N}$ such that there are pairwise distinct points $x_1$, $y_1$, $x_2$, $y_2$, $\ldots$ in $\mathbb{R}^d$ such that (i) $\left\|x_i-y_i\right\|_2 >1$ for ...
0
votes
1answer
35 views

Constructing a parallelogram according to the given condition

The question #To prove two angles are equal when some angles are supplementary in a parallelogram has been solved. In the process of solving it, I found it is not that easy to draw the corresponding ...
1
vote
1answer
18 views

Set of discrete orbits under subgroup of $Isom(\mathbb{E}^n)$ is clopen

I'm trying to solve the following exercise (exercise 1.4 from Szczepanski's "Geometry of Crystallographic Groups"): Let $\Gamma$ be a subgroup of $I(\mathbb{E}^n)$, the group of isometries on ...
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0answers
27 views

How can I show that symmetry groups of two regular n-gons are conjugate?

Let $P_1$ and $P_2$ be regular n-gons in $E^2$ with centers $C_1$ and $C_2$. Prove that $Sym(P_1)$ and $Sym(P_2)$ are conjugate in $Isom(E^2)$. Hi, I have to prove this but it looks very obvious to ...
0
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0answers
19 views

How to show the equivalence of an Euclidean norm to Euclidean distance?

I have the following problem: I have a set of $N$ vectors. I have some a priori information where the information for each vector comes from. If the two vectors $x_1$ and $x_2$ have been sampled ...
3
votes
1answer
48 views

Rotation of complex numbers in a complex plane. Check my work?

Say that $c_1 = -i$ and $c_2 = 3$. For this problem, let $z_0$ be an arbitrary complex number. We can rotate $z_0$ around $c_1$ by $\pi/4$ counterclockwise to get $z_1$. Next, we canrotate $z_1$ ...
5
votes
1answer
89 views

Could Euclid have proven that multiplication of real numbers distributes over addition?

In Euclid's day, the modern notion of real number did not exist; Euclid did not believe that the length of a line segment was a quantity measurable by number. But he did think it made sense to talk ...
0
votes
2answers
44 views

Find value of $t$ between the difference of 3D vectors.

Hint: The distance between $2$ vectors equals the magnitude of their difference. What is the value of $t$ for which the vector $\mathbf v = \begin{pmatrix} 2 \\ -3 \\ -3 \end{pmatrix} + ...
7
votes
1answer
117 views

Could Euclid have proven that real number multiplication is commutative?

In Euclid's day, the modern notion of real number did not exist; Euclid did not believe that the length of a line segment was a quantity measurable by number. But he did think it made sense to talk ...
3
votes
2answers
61 views

Distance involving 3D lines and vectors.

In this problem, a = \begin{pmatrix} 5 \\ -3 \\ -4 \end{pmatrix} and b = \begin{pmatrix} -11 \\ 1 \\ 28 \end{pmatrix} Vectors p and d exist such that the line containing a and b can be expressed in ...
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vote
2answers
17 views

3D line in a 3D plane. Find the intersection of the two.

(I'm new to Math.StackExchange, so if you see any errors, please comment below!) $\mathcal{P}$ is the plane containing the three points $(-3,4,-2)$, $(1,4,0)$, and $(3,2,-1)$. $\ell$ is the line ...
0
votes
1answer
89 views

If AC and BC are two equal chords, BA is produced to P and CP cuts the circle at T the how is CT:CB=CA:CP?

I've been solving the following question, If AC and BC are two equal chords of a circle. BA is produced to any point P and CP, when joined cuts the circle at T then show that ...
0
votes
1answer
43 views

Other way to show $ Q, H, M $ are collinear without $BECH$ is a parallelogram(IMO 2015 Problem 3 )

IMO 2015 Problem 3 Let $ABC$ be an acute triangle with $AB\gt AC$. Let $\Gamma$ be its circumcircle, $H$ its orthocenter, and $F$ the foot of the altitude from $A$. Let $M$ be the midpoint of $BC$. ...
0
votes
1answer
50 views

How to prove the External Bisector Theorem by dropping perpendiculars from a triangle's vertices?

I've found two different methods to prove Internal Angle Bisector Theorem, viz. Wikipedia ("Proof 2") method and AskMath.com method. How can we prove External Angle Bisector Theorem with ...