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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7
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2answers
539 views

Geometry IMO 1988

(IMO 1988/1) Consider two circles of radii $R$ and $r$ $(R > r)$ with the same center. Let $P$ be a fixed point on the smaller circle and $B$ a variable point on the larger circle. The line $BP$ ...
9
votes
1answer
1k views

Is there a value for $\pi$ that relates to triangles?

So I heard that if one inscribes the largest circle that can fit into a equilateral triangle, then divides the perimeter of the triangle by the diameter of the inscribed circle, it gives a value which ...
1
vote
1answer
319 views

Error Metric which incorporates both mean & standard deviation of data in euclidean space

For simplicities sake (the actually problem is more complex)...Let say I have a set of n 3d points, whose position move over time. For all pairs, I have calculated the mean and standard deviation of ...
3
votes
2answers
508 views

Inscribed quadrilateral

I need a hint on this problem. ABCD is inscribed quadrilateral. Diagonals AC and BD intersect at point O. OP and OQ are the perpendiculars from O to BC and AD. M and N are the midpoints of AB and CD. ...
0
votes
1answer
116 views

The Nearest Points

Given a set R of N points R={(x1,y1,z1),(x2,y2,z2),.....,(xn,yn,zn)} and set S of M points S={ ((a1,b1,c1),(a2,b2,c2),...(am,bm,cm))}. for each point pi(i=1 to N) in Set R ,find the point qj(j=1 to ...
8
votes
4answers
5k views

Find the area of overlap of two triangles

Suppose we are given two triangles $ABC$ and $DEF$. We can assume nothing about them other than that they are in the same plane. The triangles may or may not overlap. I want to algorithmically ...
4
votes
1answer
115 views

Three points on a line

Given $\triangle ABC$. On the side AB externally is constructed square ABPQ. On the side AC internally is constructed square ACMN. AH is the altitude. If $O_1$ and$O_2$ are the centers of the two ...
1
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2answers
62 views

rates of motion of projected points along a circle

Have I forgotten all my secondary-school geometry? (That's not actually my question.) Suppose $R>r>0$ and consider this circle (later edit: I think $R>0$, $r>0$ is enough; we don't ...
2
votes
3answers
230 views

Area and circles

I need help with this hard geometry problem. Given a $\triangle ABC$. M is the midpoint of AB. $k_1$ and $k_2$ are the circles with diameters AC and BC respectively. $A_1$ and $A_2$ are the midpoints ...
5
votes
2answers
483 views

condition for a set to be compact and convex

Is it true that a set in, say, $n$-dimensional Euclidean space is compact and convex iff its intersection with any line is empty, a single point, or a closed line segment?
4
votes
2answers
202 views

Refraction equation, quartic equation

Given two points $P$ and $Q$, a line ($A$, $B$ - orthogonal projection of $P$, $Q$ onto the line) and a coefficient $n$, I want to find out such point $C$ that $\frac{\sin{a}}{\sin{b}}=n$ (in fact, ...
0
votes
1answer
93 views

Formula rendering a triangle based off of radius and center?

Let's say I have a triangle (just a basic equilateral). What I'd like to do is render it by specifying its radius and center, and then calculate the vertices from there. What is the formula for this? ...
1
vote
1answer
5k views

How to find the interior angle of an irregular pentagon or polygon?

I have 5 points and measures of sides of pentagon in 2D. Then how do i find interior angles of pentagon? Suppose $P_1,P_2,P_3,P_4,P_5$ are five points of Pentagon $P_1P_2P_3P_4P_5$. I know how to ...
2
votes
3answers
1k views

Is there a uniform way to define angle bisectors using vectors?

Look at the left figure. $x_1$ and $x_2$ are two vectors with the same length (norm). Then $x_1+x_2$ is along the bisector of the angle subtended by $x_1$ and $x_2$. But look at the upper right ...
6
votes
1answer
333 views

A cute geometry problem about angle trisectors.

Here is a cute geometry problem I saw some time ago. I know the solution, I just wanted to share ;-) (Please, don't be mad at me.) Consider an acute triangle $\triangle ABC$. Let $AP$, $AQ$ and ...
0
votes
1answer
271 views

Relatively simple geometry with latitude/longitude coordinates

I'm trying to calculate a user's deviation from a route defined as a list of latitude/longitude points that make up the route. My proposed solution is to try and calculate the distance between the ...
1
vote
1answer
107 views

Area and orthocenter

I need help with the following proble. Given a triangle ABC with orthocenter H and altitudes $CM= 2 \sqrt 2$ and $AN=3$. If H divides the altitude BP into segments with ratio 5:1, i.e.BH:HP=5:1, ...
-1
votes
4answers
678 views

Why are there isosceles triangles? [closed]

Why are isosceles triangles called that — or called anything? Why is their class given a name? Why did they find their way into the Elements and every single elementary geometry text and course ...
1
vote
2answers
86 views

How to show that if $P_1,P_2\in\mathbb{R}^3$ then the straight line connecting them is the shortest one?

Let $P_1,P_2\in\mathbb{R}^3$ and consider all the paths from $P_1$ to $P_2$, I wish to prove that the euclidean distance (that is the length of the line connecting them) is the distanse of the ...
2
votes
0answers
410 views

Euler's Line of a medial triangle

I have the following problem with a comment below on the steps that I took so far. Here is the example: Let triangle ABC be any triangle. The midpoints of the sides in Triangle ABC are labeled $A', ...
1
vote
3answers
71 views

Represent lengths rectangle using given terms

In a rectangle, $GHIJ$, where $E$ is on $GH$ and $F$ is on $JI$ in such a way that $GEIF$ form a rhombus. Determine the following: $1)$ $x=FI$ in terms of $a=GH$ and $b=HI$ and $2)$calculate $y=EF$ in ...
2
votes
2answers
539 views

Prove there exists a triangle without using Euclidean Parallel Postulate

Let $a$ and $b$ be real numbers where $0 < a< b<180$. Let $A$, $B$, $D$ be points so $A$-$B$-$D$. Part 1: Prove there exists a triangle $ABC$ where measure of angle $CAB$ is $a$ and measure ...
1
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1answer
154 views

Similar triangles

Knowing that Triangle $LAB$ is similar to Triangle $LRQ$, prove that the length of $QR$ is constant while point $L$ varies. There are two circles intersect at points $A$ and $B$. $L$ is a point on ...
11
votes
1answer
430 views

A hard problem from regular $n$-gon

I found the following theorem in a book of mine without a proof. Could someone show me a proof of it? Given a regular $n$-gon, with $n$ odd and vertices $v_1,\ldots,v_n$, and $C$ its circumcircle. At ...
0
votes
1answer
82 views

Getting as close to a target vector as possible

Say I have P number of political parties in an election that I'm trying to rig. My boss has decided what number of percentage of the votes it is best that each party should get. N number of votes have ...
0
votes
2answers
192 views

velocity confusion

A velocity encompasses both speed and direction in a single vector. I'm a little bit confused about how to separate the two. I have 2 creatures. The first is located at position (x1, y1). The second ...
0
votes
1answer
734 views

Prove that intersection of diagonals in trapezium divide parallel segment to equal segments

I've got exercise to do as en exercise to my school leaving exam and I have no idea how to prove it: Diagonals of trapezium intersect in point $S$. Through point $S$ the segment was given that ...
0
votes
1answer
151 views

Geometrical construction of the product on $\mathbb R$ [duplicate]

Possible Duplicate: Representing the multiplication of two numbers on the real line Consider the real line in the plane. Suppose you are given the location of the point associated to $0$ ...
7
votes
1answer
189 views

A scalar product in the space of oriented volumes?

Let $L\colon \mathbb{R}^n \to \mathbb{R}^N$ be an injective linear map. By the Cauchy-Binet formula, $\det(L^TL)$ equals the sum of the squares of all minors of $L$ of order $n$: this looks just like ...
1
vote
0answers
360 views

Prove a very original version of Descartes's circle theorem

Prove: I define the radius of three mutually externally tangent to be $d,e,f$ respectively. The circle with radius $x$ is internally tangent to all three circles. Then $$ddeeff+ddeexx+ddffxx+eeffxx ...
3
votes
2answers
215 views

Stuck on geometry proving

Points S and D are respectively the center of the circle circumscribed on the acute triangle ABC and the orthocenter of this triangle. Prove that ASBX, where X is the center of the circle ...
2
votes
1answer
76 views

Normal to a plane

My textbook makes a quick, unproven claim that if a vector P is orthogonal to /one/ vector that lies on a plane G, then P is normal to the plane. My question is, is this a mistake? Wouldn't you need P ...
3
votes
1answer
118 views

Determinant and Measure

The determinant of the matrix of its vectors gives the measure of an $n$-dimensional parallelogram. For example, in $2$ dimensions, the area spanned by vectors $v$ and $w$ is \begin{array}{|cc|} v_1 ...
1
vote
2answers
300 views

Can more than four circles internally tangent or external tangent or combination of both each others at different points?

Is it true for infinite number of m, more than four, there exist m circles internally tangent or external tangent or combination of both each others(in this problem, i mean a circle must be tangent to ...
18
votes
2answers
2k views

Intersection of two parabolae

Problem: Consider two parabolae such that their axes of symmetry form a right angle. Prove that all four points of intersection lie on a common circle (it is an assumption that there exist such four ...
2
votes
2answers
2k views

The sides of an isosceles triangle from the circumradius and inradius

I need to solve the following problem only by using Pythagoras Theorem and congruent triangles. Find the sides of an isosceles triangle ABC with circumradius R=25 and inradius r=12.
2
votes
2answers
622 views

Prove that two sides are parallel in the reflection of an isosceles triangle

Given an isosceles triangle $\triangle{BAC}$ as the one in the figure below and the reflection in line $l_{BC}$ that transforms the triangle into $\triangle{B'A'C'}$. How can I prove that $s_{AB} ...
2
votes
1answer
2k views

how best to draw two planes intersecting at an angle which isn't $\pi /2$?

What's the best way to draw two planes intersecting at an angle that isn't $\pi /2$? If I make them both vertical and vary the angle between them, the diagram always looks as though our viewpoint has ...
0
votes
2answers
48 views

Inner product in $\mathbb{R}^2$ and angles of a triangle

Let $P_1,P_2,P_3$ be $3$ different points in $\mathbb{R}$, then $P_1,P_2,P_3$ form a triangle. What is the relation between the (one of the) angles of this triangle and $\langle P_2-P_1,P_3-P_1 ...
4
votes
0answers
101 views

Unlit region in a Room lined with Mirror

Mathematician Ernst Straus wondered if a room lined with mirror can always be lit with a single match. He (or someone else) discovered that in the following room, light shone from A can't reach B: ...
9
votes
1answer
286 views

Name of a Euclidean Geometry Theorem

There is a well known theorem in plane Euclidean geometry as follows, and I would like to know only the name by which it is known. Theorem: Let $ABC$ be a triangle. Choose points $D,E,F$ on the ...
2
votes
2answers
2k views

Convex Quadrilateral Test

I have a four points in plane and need to test (based on point coordinates) whether they are able to form a convex quadrilateral: Of course, the test should avoid configurations like these: ...
4
votes
1answer
198 views

Why do derivatives of certain equations relating to circles yield other similar equations? [duplicate]

Possible Duplicate: Why is the derivative of a circle's area its perimeter (and similarly for spheres)? We all know that the volume of a sphere is: $V = \frac{4}{3}\pi r^{3}$ and its ...
2
votes
1answer
84 views

Discriminant of the Coxeter group $I_2(n)$.

Coxeter group $I_2(n)$ is just a dihedral group $D_{2n}$. This group acts on the plane $\mathbb{R}^2$ and after complexification on $\mathbb{C}^2$, thus it acts on $\mathbb{C}[z_1, z_2]$. Ring of ...
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votes
2answers
110 views

Problem related “polygon offseting”

Knowing points P1,P2,P3 and distance d and the angles shown in the figure, angle between a and b not necessary 90º What's the size of K?
3
votes
1answer
455 views

Computing Hermite Normal Form using Extended Euclidean Algorithm

I am trying to calculate the Hermite Normal Form of a square $n \times n$ matrix using the Extended Euclidean Algorithm to compute the columns of the HNF matrix, rather than the standard (column) ...
11
votes
1answer
569 views

Sub-determinants of an orthogonal matrix

Let $A$ be a matrix in the special orthogonal group, $A \in SO_n$. This means that $A$ is real, $n \times n$, $A^t A = I$ and $Det(A)=1$, that is, the column vectors of $A$ make a positively-oriented ...
2
votes
0answers
71 views

Plane tessellation $6^2*3^2$

An article I am reading mentioned "the plane tessellation $6^2*3^2$", I tried looking it up and I found all sort of plane tessellations - but not $6^2*3^2$. However, I did find information about ...
0
votes
1answer
954 views

Making a circle with paper folding, scissors, pencil, and a straightedge

Can we make a circle using paper folding, scissors, straightedge, anda pencil, allowing an infinite number of operations? I think my chemistry teacher have show me once how to make it during the ...
6
votes
2answers
131 views

The role of string in constructive geometry

I was wondering whether, if I add string and thumbtacks to my geometry kit, I am able to do any new constructions. The idea being, with string, I can draw ellipses for instance and the intersection of ...