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.

learn more… | top users | synonyms (1)

0
votes
1answer
208 views

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 ...
0
votes
1answer
73 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 ...
4
votes
2answers
182 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 $BC,CA,AB$ ...
3
votes
1answer
167 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
votes
1answer
50 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 ...
2
votes
4answers
102 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 $DF^2=2\cdot AE^2$.
6
votes
1answer
91 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 $p$ ...
2
votes
1answer
199 views

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 ...
1
vote
1answer
88 views

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 to ...
10
votes
3answers
853 views

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 ...
1
vote
0answers
134 views

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 ...
0
votes
2answers
111 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 \...
0
votes
1answer
60 views

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 ...
0
votes
2answers
62 views

Find the value of $a$.

please help I'm lost on what numbers to add or what formula to use
0
votes
1answer
760 views

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?
2
votes
2answers
246 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 ...
1
vote
1answer
93 views

Is there anything we can add to the present Euclidean Geometry?

I did not study the detail history of Euclid. As far as I know, the geometry starts from a few postulates and keeps on expanding (through theorems developing) to the present state. Even with these ...
1
vote
2answers
243 views

Constructing two tangents to the given circle from the point A not on it

I'm trying to complete Level 21 from euclid the game: http://euclidthegame.com/Level21/ The goal is to construct two tangents to the given circle from the point A not on it. So far I've figured ...
1
vote
0answers
37 views

RFC: Proof relating area to intersection of points.

this is the first time I used latex. Please excuse the rough edges. I asked a question about this problem here: Relating area to a line intersecting with a point. but I think I was able to find a ...
1
vote
1answer
145 views

Can any vertex of an isosceles triangle represent the centre of a circle, and the base vertices represent points on the circumference of that circle?

This question occurred to me doing this circle geometry problem, and I was wondering if anyone could clear it up. Geometrically, it seems it would make sense, provided that 2 sides are equal (equal ...
3
votes
1answer
410 views

Distance between two barycentric coordinates

I am developing a system, and generally in this system we examine the effect of a number of factors on our data. We choose to use Barycentric coordinates to help us to achieve that. I have no problem ...
2
votes
2answers
199 views

Simple 9-th grade geometry problem

I have a geometry problem which states that Find the range of $x$ in following figure. Given that $AD$ and $AC$ are equal, and the values and angles are also given. How to estimate the range ...
1
vote
1answer
393 views

dot product of vectors with not orthogonal basis

The dot produt (inner product in the context of Euclidean space) of two vectors $\mathbf{a}=\left [ a_{1},a_{2},...,a_{n} \right]$ and $\mathbf{b}=\left [ b_{1},b_{2},...,b_{n} \right ]$ is defined as:...
1
vote
1answer
56 views

Relating area to a line intersecting with a point.

I really could use a hint with this following problem: If a line L separates a parallelogram into two regions of equal areas, then L contains the point of intersection of the diagonals of the ...
1
vote
1answer
506 views

Incentre and excentre of a triangle

Prove that the triangle formed by the points of contact of the sides of a given triangle with the excircles corresponding to these sides is equivalent to the triangle formed by the points of contact ...
0
votes
3answers
104 views

Equation of rectangle

I need equation of a rectangle on the Cartesian coordinate system. Is there an equation for a rectangle? for example equation of ellipse is $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$
0
votes
1answer
22 views

Solving the following relation in triangle

If a line through the centroid $G$ of a triangle ABC meets $AB$ in $M$ and $AC$ on $N$ then prove that $AN. MB+AM. NC=AM. AN$ both in magnitude as well as sign I tired dividing the equation by $AM. ...
1
vote
1answer
39 views

Function on plane with incenter

Let $f$ be a function from the set of all points on the plane to the nonzero real numbers. Suppose that for any triangle $ABC$ with incenter $I$, we have that $f(I)=f(A)f(B)f(C)$. What are the ...
0
votes
1answer
51 views

Doubts about locus and its equation

Two points A and B with $(1,1)$ and $(-2,3)$ respectively are given.find the locus of point P.So that area of $\Delta$PAB is 9 square units. answer is :- $2x+3y+13=0$ or $2x+3y-23=0$. how i tried:- i ...
1
vote
2answers
239 views

Geometry : find the points of tangency between two lines and two circles [closed]

I have a programming problem. I need to find the intersection points between two lines tangent to two circles and the circles! I have the circles' radiuses and centers. So I need points ...
3
votes
3answers
588 views

Average distance from a point in a ball to a point on its boundary

What variety of methods are readily available to find the average distance from a point in $\{ (x,y,z) : x^2+y^2+z^2 \le r^2 \}$ to the point $(0,0,r)$? I just worked this out and got $6r/5$. Later ...
2
votes
1answer
108 views

How to embed this circle tangent to the other circles?

I want to construct a circle that would be tangent to the $3$ circles and would have its diameter lie somewhere on the segment $BI$. $EF$ includes the diameters of the $3$ given circles. $EB=BF$. ...
4
votes
2answers
142 views

What is the ratio of the side length of a regular hepatgon to the side length of the internal heptagon?

Given a regular heptagon with side length 1, create a star heptagon by connecting every vertice. Note that removing the "points" of the star yields a similar heptagon. I want to know the side ...
4
votes
1answer
106 views

Circle with perpendicular chords

A blue circle is divided into $100$ arcs by $100$ red points such that the lengths of the arcs are the positive integers from $1$ to $100$ in an arbitrary order. Prove that there exists two ...
3
votes
3answers
224 views

How can I find the volume of prism: $V = \frac{(a + b + c)Q}{3} $

In the book Handbook of Mathematics (I. N. Bronshtein, pg 194), we have without proof. If the bases of a triangular prism are not parallel (see figure) to each other we can calculate its volume by the ...
1
vote
3answers
66 views

Show a complex equation has one or two roots

Let $a$ $\neq$ $0$, $b,$ and $c$ be complex constants. Show that the quadratic equation $az^2+bz+c=0$ has one or two roots. My thoughts: Let $a=a_1+ia_2,$ $b=b_1+ib_2,$ and $c=c_1+ic_2$. I also ...
1
vote
1answer
58 views

To use Vieta's formula for complex constant solution or not?

Let $b$ and $c$ be complex constants such that $z^2$ + $bz$ + $c$ = $0$ has two different real roots. Show $b$ and $c$ are real. I think I need to be using Vieta's formula, however I have solved it ...
-1
votes
1answer
230 views

The point of contact between a line with a circle

My question is: I have a circle of radius 40 and a line which the circle is tangent to. So, if I take a circle of radius 80, do the two circles have the same point of contact? I mean: do they (my ...
1
vote
2answers
2k views

How do 3 points define a plane?

I was solving a combinatorics problem which asked me to find the number of planes that can be constructed from a set of 25 points such that no 4 points in the set of 25 points are co-planar and then I ...
-1
votes
2answers
795 views

Is it appropriate to apply Euclidean Distance to Complex Numbers?

Would complex numbers be considered as part of Euclidean Space? Would this measurement give an accurate result? If not, what would be a more appropriate distance measurement/similarity measure with ...
3
votes
1answer
45 views

Is for open connected $U$ the set $U_\varepsilon$ for small $\varepsilon$ connected?

Let for an open connected subset $U\subset \mathbb R^n$ and a number $\varepsilon >0$: $$ U_\varepsilon=\{x\in U: dist (x, \partial U)> \varepsilon \}. $$ Then $U_\varepsilon$ is open but in ...
0
votes
1answer
64 views

Roundness in Taxicab Geometry

I was just wondering whether circles are considered "round" still in taxicab geometry. I know that "roundness" is probably not a well-defined term and I know what a circle /appears/ to look like in ...
2
votes
3answers
70 views

Euclidean “straight line” calculation

Please see image first.. I have as input the following (I presume these are in effect Euclidean coordinates): The angle and the length of the red line. The angle and the length of the green line. ...
0
votes
1answer
50 views

Unique Euclidean isometry between affinely independent points

Let $u_0,\dots,u_n$ be vectors in $\mathbb{R}^n$ such that $u_1-u_0,\dots,u_n-u_0$ are linearly independent and similarly let $v_0,\dots,v_n$ be vectors in $\mathbb{R}^n$ such that $v_1-v_0,\dots,v_n-...
0
votes
2answers
125 views

Derive formula for coordinates of internal and external centers of similitude.

Given 2 circles $(x - x_1)^2 + (y - y_1)^2 = r^2$ and $(x - x_2)^2 + (y - y_2)^2 = r'^2$ (with radii $r, r'$) coordinates of the internal and external centers of similitude $C_i, C_e$ are given by $...
0
votes
4answers
75 views

Complex Numbers: Im$(\frac{12}{z-7})=1$

Sketch and describe the set of complex numbers satisfying $$Im(\frac{12}{z-7})=1$$ where $z=x+iy$ The answer should be in circle form. Here is what I have so far: $$Im(12)=z-7$$ $$Im(12)=x+iy-7$$ ...
1
vote
0answers
67 views

Banach Tarski paradox in Minkowski space

The Banach - Tarski paradox states: 'A three-dimensional Euclidean ball is equidecomposable with two copies of itself.' My simple question is: Does this paradox hold his validity also in the Minkowski ...
5
votes
1answer
1k views

IMO 2014 problem 3, first day

Convex quadrilateral $ABCD$ has $\angle ABC = \angle CDA = 90^{\circ}$. Point $H$ is the foot of the perpendicular from $A$ to $BD$. Points $S$ and $T$ lie on sides $AB$ and $AD$, respectively, such ...
4
votes
4answers
3k views

How to make a perpendicular construction in 3 moves?

I've been playing Euclid: The Game for some time now. I'm quite addicited to it, trying to get all the records now. Suprisingly, I'm not able to get a record for some really early level. In Level 4 (...
0
votes
0answers
88 views

A mixtilinear tangency

Let ABC be a triangle with incircle $\gamma$ and circumcircle $\Gamma$. Let $\Omega$ be the circle tangent to rays $AB, AC,$ and to $\Gamma$ externally, and let $A^{\prime}$ be the tangency point of $\...