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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Kiselev's Book I Plainimetry Question 242 - Question in the Description

Two lines passing through a point Μ are tangent to a circle at the points A and B. Through a point С taken on the smaller of the arcs AB, a third tangent is drawn up to its intersection points ...
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1answer
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An inequality about the areas of two triangles

There is point $P$ in a triangle $ABC$. $Q,R,S$ are the symmetric of $P$ with respect to the sides $AB,BC,CA$ respectively. I have to prove that the area of $ABC$ is $\geq$ than the area of $QRS$. ...
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Configurations of eleven (or more) points in the Euclidean plane, such that out of any four there is a pair at unit distance.

Inspired by this question, I was wondering the following: What is the maximal size of a subset $C$ of the Euclidean plane such that out of any four points in $C$ there are two at unit distance ...
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Elementary geometry question involving quadrilateral and bisector given with picture.

$ABCD$ is a quadrilateral. $m(\widehat{BAC})=48^\circ$. $m(\widehat{CAD})=66^\circ$. $m(\widehat{CBD})=m(\widehat{DBA})$. What is $\color{magenta}{m(\widehat{BDC})=x}$? Tried lots ...
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2answers
65 views

10 points outside a unit circle

Let $P_1$, $P_2$,... $P_{10}$ be ten points outside the unit circle centered at the origin $O$. Given that $\|P_iP_j\|\ge 1/\sqrt{2}$ for all $1\le 1<j\le 10$, find the minimum of the sum of the ...
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What is the perspective projection of a 3d point relative to a quarternion encoded camera?

I'm representing a camera on the cartesian space as a tuple of a 3d point (position) and a quarternion (rotation). I get the front, right and up vectors of the camera by applying the quaternion to the ...
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About the stable/invariant point sets in a plane with respect to shift/linear transformation

I'm reading Vlademir A. Zorich's Mathmatical Analysis I, meeting exercise question as following: a) A set $S \subset X$ is stable with respect to a mapping $f:X \rightarrow X$ if $f(S) \subset ...
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construct a triangle with a compass and a ruler, given $a, B, t_a$

$a,b,c$ the sides of the triangles; $A,B,C$ the angles of the triangles; $t_a, t_b, t_c$ the internal bisectors of the angles $A,B,C$. How to construct a triangle with a compass and a ruler(a ...
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233 views

Hexagon packing in a circle

Suppose I want to pack hexagons in a circle, as on the drawing below (red indicates "packed" hexagons). I am wondering what is known about this problem. Specifically, I am interested in an ...
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1answer
54 views

Let the plane V be defined by $ax + by + cz + d = 0$; with $a, b, c, d \in \mathbb{R}$ and the vector $(a; b; c)$ a unit vector.

I am battling to get my mind around some of the concepts involving vectors in $3$-space. This question asks me whether the following statements are True or False: (A) The line $(a; b; c)$ is parallel ...
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422 views

Which Area of mathematics can explain this?

http://i.stack.imgur.com/rij3X.png As in the image we can see that ray of light is bouncing off objects. Black ones are opaque objects and white ones are transparent objects. I want to calculate how ...
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2answers
413 views

Bounding box enclosing circles, that complies with ratio constraints

Given a circle centered at $A$, with radius $R_a$ and another radius $R_b$, I need to find a center for circle $B$ such that both circles are tangential, and the bounding box including both circles ...
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About Euclid's Elements and modern video games

Update (6/19/2014) $\;$ Just wanted to say that this idea that I posted more than a year ago, has now become reality at: http://euclidthegame.com/ 12.292 users have played it in 96 different ...
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1answer
22 views

Expression of reflection isometry in the complex plane

Using the fact that an anti-displacement in the plan has the form $$f(z) = a \overline{z} + b$$ I have done some computation to find the reflection about the line passing through two points $P$ and ...
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Neccesary condition for perpendicularity [on hold]

Triangle $\mathit{BCD}$ lies in plane $P$ and $\mathit{AD}$ and $\mathit{DC}$ are perpendicular ($A$ is the top of the prism $\mathit{ABCD}$) $\mathit{AD}$ is perpendicular to $P$ if which of the ...
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1answer
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1answer
66 views

Manifold that is not a Euclidean space

I just started reading a textbook, and it keeps saying that an $n$-dimensional manifold is a topological space with the same local properties as Euclidean $n$-space. I don't really understand what is ...
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2answers
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$AB=AC$, $BD$ is the angle bisector of $\angle B$ , find $\angle A$

Let $ABC$ be an triangle, $AB=AC$. $BD$ is the angle bisector of $\angle B$, $BD$ intersect $AC$ at point $D$, and $AD=BC+BD$. show that: $\angle BAC=20^\circ$ Well, If $\angle BAC=20^\circ$, I ...
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64 views

Trapezoids in a square

Good day As part of a problem I need to show that AB is parallel to CD, with the given info on the image. All the segments marked red are equal, all 1-stripe grey equal etc. I'd like to prove ...
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1answer
43 views

Geometry - Trapezium and its properties

In the trapezium $ABCD$ , $AB$ is parallel to $CD$ and $O$ is the intersection of $AD$ and $BC$. The line $PS$ is drawn through $O$ in such a way that $PS$ is parallel to $DC$. If $AB = 20$ AND $CD = ...
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Easy question, hard solution: find the area about a domain in the plane?

We want to find the area of a domain with piecewisely smooth boundary by using the coordinates $(p,\theta)$ of the random line: It has been known that every straight line $\ell$ on $R^2$ can be ...
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1answer
39 views

A rectangle with a triangle

ABCD is a rectangle. From 'C' two lines are drawn to meet AB and AD at E and F respectively (here AB and AD are not produced). From B, a line is drawn to meet CE, DE and AD at G, H, and F ...
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Subgroup of the orthogonal group generated by reflections

What is the subgroup of the orthogonal group $O(n)$ generated by relections around $k$-dimensional hyperplanes? For $k=n-1$ it is $O(n)$ by the Cartan's theorem. For $k=0$ it is $\{\pm Id \}$. What ...
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calculating volume of a horizontal cylindrical tank from depth

Has anyone found a good formula to convert a depth of fluid to a volume remaining in a tank for a cylindrical tank laying horizontal? (with or without half sphere end caps) Much Thanks!
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A question on tangent circles and finding the angle between the lines

Two circles of radii in the ratio $1:2$ touch each other externally. The center of the small circle is '$c$' and that of the bigger circle is '$D$'.The point of contact is $A$. $\overline{PAQ}$ is a ...
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1answer
72 views

2011 USAMO Problem 3, Hexagons.

In hexagon $ABCDEF$, which is nonconvex but not self-intersecting, no pair of opposite sides are parallel. The internal angles satisfy $\angle A = 3\angle D$, $\angle C = 3 \angle F$, and $\angle E ...
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Collinearity problem (Newton-Gauss line)

I had some troubles with this problem : Let $ABCD$ be a convex quadrilateral. $M$ and $N$ are the midpoints of the diagonals $AC$ and $BD$. The sides $AB$ and $CD$ are extended until they ...
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31 views

Triangle inequality $ax ≥ br + cq$

I got stuck on this problem : Given a triangle (△ABC) of sides $a$, $b$ and $c$, let $O$ be a point inside △ABC. Let $D$, $E$ and $F$ be points on sides a, b and respectively c such that $OE ⊥ ...
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3answers
40 views

Triangle area inequalities

I've got stuck on this problem : Proof that for every triangle of sides $a$, $b$ and $c$ and area $S$, the following inequalities are true : $4S \le a^2 + b^2$ $4S \le b^2 + c^2$ ...
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488 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 ...
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215 views

Which is the correct Hessian matrix (the standard matrix of a bilinear form)?

Please note the typo in the first entry: $\frac {\partial^2f} {\partial x_1\partial x_2}$ should instead be $\frac{\partial^2f} {\partial x_1\partial x_1}$. Also, this Hessian matrix need not be ...
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470 views

Prove the similarity of isosceles triangles…

Two similar isosceles triangles are constructed outside of an parallelogram ABCD, the first being $ABB_1$ and second $CBC_1$ i.e. $|AB| = |AB_1|$ and $|CB| = |CC_1|$. Since $ABB_1$ and $CBC_1$ are ...
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Non-orthogonal space

What does the angle between two non-orthogonal basis denote? Is it correlation or some measure of dependence. Does that mean that coordinates of a point if moved in the direction of one axis also ...
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1answer
32 views

$S \leq \frac{(a+b)(c+d)}4 $

I got stuck on this problem: Given a convex quadrilateral of area $S$ and sides $a$, $b$, $c$ and $d$, prove that: $$S \leq \frac{(a+b)(c+d)}4$$ What I've done so far was to proof that ...
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Proof of Menelaus using areas

I've tried to proof Menelaus' theorem using areas, but I've didn't figure out how. Some suggestions would be appreciated. Menelaus' Theorem states : Given a triangle ABC and a transversal ...
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79 views

Geometry - angle bisector, circumcircle: SL olympiad

I tried this problem as much as I can, but I got nothing. This is a Sri Lankan mathematical olympiad problem. Let $P$,$Q$ be points on the sides $AB$ and $AC$, respectively, of a $\triangle ABC$ ...
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81 views

Extension of metric definition to two sets

The standard definition of a metric, is a function $d: X\times X \to \mathbb{R}$. What is a sensible/common extension of a metric/pseudometric to $\tilde d: X\times Y \to \mathbb{R}$, i.e. distance ...
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524 views

Find locus of points relating to an ellipse

I would like to find the equation of the following locus. For a big circle C centered at (0,0), the locus of points that the sum of distances to Y-axis and to C is 1, say in the first quadrant, is ...
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2answers
43 views

Quadrilateral's area problem

I have some troubles with this problem : Let $ABCD$ be a convex quadrilateral. $M$, $N$, $P$ and $Q$ are the midpoints of the sides $AB$, $BC$, $CD$ and $AD$. $AN$, $BP$, $MD$ and $CQ$ are ...
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3answers
93 views

The sum of squares of two line segments formed by a circle and coordinate axes

We have an angle of 90° so that there are 2 points A, B on each side of the angle, O is the vertex and |OA| = |OB|. On the arc AB with it's center being in O, we pick an arbitrary point P and draw ...
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3answers
279 views

Eritrea's Theorem

According to this newspaper, an Eritrean high school student named Saied Mohammed Ali has discovered a new geometric theorem. Another source seems to say that it's the following: Say you have a ...
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1answer
387 views

Intuition - The Shortest Curve Between Two Points is a Line

The references below aver that the following is not crudely trivial: The shortest curve between two points is a (straight) line. An elementary school teacher construed it as follows ...
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Given two rectangles A and B and their dimensions, is a there test for lengthA<lengthB, widthA<widthB [closed]

If we have two rectangles and their dimensions, is there a mathematical test to simultaneously compare the two numbers. For examples, if we wanted to compare the area, we would compare the products, ...
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Finding tangents to a circle with a straightedge

There is a geometric construction that I heard years ago and I still haven't figured out why it works despite several attempts. Playing with pen, paper and GeoGebra makes me confident that it does ...
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1answer
330 views

Do collinear lines or overlapping collinear line segments intersect?

I am writing a function to find the intersection of a pair of lines and another function to find the intersection of a pair of line segments. The parallel case and the single point intersection case ...
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3answers
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: ...
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Surface Area of a Self-Intersecting Spindle Torus in Flat Euclidean space

Does the usual formula for a torus from Pappus's theorem apply to self-intersecting spindle tori? To avoid any ambiguity in defining a notion of a surface area for the self-intersecting torus - I ...
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1answer
1k views

How to plot N points on the surface of a D-dimensional sphere roughly equidistant apart?

Let's say I have a D-dimensional sphere with a radius R. I want to plot N number of points evenly distributed (equidistant apart from each other) on the surface of the sphere. It doesn't matter where ...
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3d parametric spiral to 3D goldean mean spiral

I know I can create a 3d parametric spiral with the formula below but How can I do the same thing with goldean spiral? I looked at https://en.wikipedia.org/wiki/Golden_spiral but I don't see how to ...
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189 views

An inequality about the sum of distances between points : same color $\le$ different colors?

When I was drawing some points on paper and studied the distances between them, I found that an inequality holds for many sets of points. Suppose that we have $2$ blue points $b_1,b_2$ and $2$ red ...