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
0answers
19 views

Two equal inscribed squares

A triangle has inscribed squares. Can an area of two of them are equal if the side which is common to the triangle and square are not equal?
2
votes
1answer
37 views

Regular polygon inside another?

Inspired by this question, I was wondering if one can generalize to the case of an $n$-gon. For example, when $n=5$ we have this picture: where $ABCDE$ is a regular pentagon, ...
0
votes
1answer
44 views

Square is a parallelogram?

I remember, in the geometry class, our teacher used to tell us some definitions or something that i don't really know about. Why is square a parallelogram?
4
votes
0answers
34 views

What's a good text to read before Coxeter's Geometry Revisited?

I am interested in reading Coxeter's famous text Geometry Revisited. It's not clear to me what the prerequisites for this text are, however. I'm sure I have enough mathematical maturity: I know ...
0
votes
0answers
14 views

Justify each step in the following proof of Proposition 3.9 (b). I like to know if I'm in the right track and I'm also missing a few of them

Proposition 3.9(b): If a ray emanates from an interior point of triangle ABC, then it intersects one of its sides. proof (a) Let r be a ray emanating from an interior point D. (b) The ray AD ...
0
votes
0answers
9 views

Parallelogram and Areas

Consider the parallelogram $ABCD$. On sides $BC$ and $CD$ take points $E$ and $F $ respectively such that $\frac{BE}{EC} = \frac{CF}{ FD}$. If the segments $AE$ and $AF$ cut $BD$ at $K$ and $L$, show ...
0
votes
0answers
21 views

Justify each step in the following proof of Proposition 3.9 (a). I like to know if I'm in the right track and I'm also missing a few of them

Proposition 3.9(a): If a ray r emanating from an exterior point of triangle ABC intersects side AB at a point between A and B, then r also intersects side AC or side BC. proof. (a) Let r= array XD ...
0
votes
1answer
19 views

Measure of angle formed by chords and two circles

The following is a question from a practice GRE Math Subject Test: In the Euclidean plane, point A is on a circle centered at point O, and O is on a circle centered at A. The circles intersect at ...
0
votes
1answer
17 views

Determine if this interpretation satisfies axiom Congruent axiom 1. If so, prove it. If not, find specific

Recall the interpretation of the rational plane: points are ordered pairs $(x, y)$ with $x, y \in \mathbb{Q}$; lines are solution sets of equations $ax + by + c = 0$ with $a, b, c \in \mathbb{Q}$ and ...
5
votes
0answers
41 views

Prove that OD is a the angle bisector of the angle BOC.

Let ABC be a non-isosceles triangle and I be the intersection of the three internal angle bisectors. Let D be a point of BC such that $ID \perp BC$ and O be a point on AD such that $IO \perp AD$. ...
0
votes
0answers
6 views

average distance between vectors of n dimenstions [on hold]

In a recent experiment I did, I observed that the minimum euclidean distance for vectors of about 10k dimensions (each single feature is has standard normal distribution) is about 20, even if I sample ...
-5
votes
1answer
68 views

On Proving that the first euclidean axiom is wrong [on hold]

Well, The first axiom in the euclidean geometry is "A straight line segment can be drawn joining any two points". But I think that there are points that can't be joined: In the image below, We have ...
1
vote
2answers
22 views

Area of triangle interior to parallelogram

Hi you can help me with this exercise? I have to find the area of the triangle $QOP$ in terms of the parallelogram $ABCDB$ but I do not know how to prove that area triangle $COD$ is $1/8$ of $ABCD$ ...
1
vote
2answers
29 views

Get canonical equation of ellipse

We have an ellipse with a circle in it. The circle is passing through the two vertices and through the ellipse's center. It's diameter equals 7. We have also an ...
0
votes
1answer
32 views

Do such triangles exist?

Are there two triangles with equal angles and two equal sides which are not congruent? I think it is impossible.
1
vote
2answers
56 views

the set of points equidistant from $ u $ and $v$ form a line.

Let $u$ and $v$ be two vectors in $ \mathbb{R}^2 $ with the standard norm. Show that the set of points equidistant from $ u $ and $v$ form a line. I show that if $x$ is equidistant from $u$ and $v$, ...
2
votes
1answer
25 views

Do two lines lay in the same 4-dimensional plane

I don't know how quite to phrase this, but I'll try. Because two point are co-linear and two lines cannot always be used to define a plane and aren't always in the same plane, are two lines always ...
8
votes
2answers
272 views

How to show that five points in ℝ³ are cospherical?

There are many conditions equivalent to the cocircularity of four points on a plane, however i could not find any such lists for the three-dimensional analog. When do five points in three-dimensional ...
11
votes
1answer
98 views

Largest $n$-vertex polyhedron that fits into a unit sphere

In two dimensions, it is not hard to see that the $n$-vertex polygon of maximum area that fits into a unit circle is the regular $n$-gon whose vertices lie on the circle: For any other vertex ...
0
votes
1answer
14 views

How could we define the existence of an object/element in the Euclidean space?

Let X be an object/element, What does it mean when I say "X is an object in the Euclidean space"? in other words, What differs an existed object from an unexisted one in the Euclidean space?
0
votes
1answer
27 views

Euclidean Geometry Construction

I am looking for an aswer to the following construction construct a triangle given two angles (3 angles) and the sum of two sides
1
vote
0answers
20 views

Incidence Geometry

Consider a quadraple $(a,b,c,d)$ of points in the real plane such that $|ab| = |cd|$. If the perpendicular bisector of line segment $ac$ is parallel to perpendicular bisector of $bd$, then how does ...
1
vote
1answer
32 views

Fundamental domains of Dihedral groups

Let $D_n$ be dihedral group of order $2n$, it acts on plane $\mathbb{R}^2$ in a standard way, by rotations and reflections. How one can find fundamental domains for such action?
0
votes
1answer
24 views

Given three concurrent lines $a,b$ and $c$, find the circunference tangent to $a$ and $b$ and with center at $c$

I have these three lines, and I need to construct a circumference tangente to two lines and that has center at the other line. I tried to construct the perpendicular lines that passes through the ...
0
votes
2answers
16 views

Given two points $A$ and $B$ and two distances $m$ and $n$, find a point that has distance $m$ fom $A$ and $n$ from $B$

I know that, as long as the distance from $|GI|<m+n$, as you can see in the figure $1$, I can constructo such point by the intersection of the circles with center at $G$ and radius $m$ and with ...
1
vote
1answer
14 views

Determine the isometric group $G$ which transfers a square into it self

I am solving the following exercise: Determine the isometric group $G$ of the euclidean plane which transfers a square into it self. The restriction of an element $g \in G$ on the vertices of ...
-5
votes
1answer
64 views

Geometry problem.

I have to find what is theta($\angle$GOE = $\angle$CDE). Here is a condition for above shape: The shape OCG is a quarter of unit circle(center is O). The line DF is a tangent line of curve CG ...
1
vote
1answer
43 views

Euclidean geometry question

Let $(P,L,\varepsilon)$ be a plane with finitely many points (i.e $P$ is finite) Assume in addition to the axioms of incidence that for each $Q \in P$ and $l \in L$ with $Q \not\varepsilon l$ there ...
0
votes
1answer
25 views

How many vectors exist satisfying the angle between any two vectors equals to a constant $\beta$ with $0<\beta<\pi$ in a $n$-dimension Euclid space?

At first, if $\beta=\pi/2$, we know that at most $n$ such vectors exist, that is, orthogonal vectors. It's obvious that the number of vectors is influenced by the angle $\beta$. Assume we've already ...
1
vote
2answers
158 views

Stupidly simple geometry problem I can't do

Okay. Here it goes. C and D are two points on the same side of a straight line AB and P is any point on AB. Show that PC + PD is least when the angles CPA and DPB are equal. I have no idea why I ...
0
votes
0answers
25 views

Determine position and orientation of a rigid object, given certain limited informations

I have a rigid 3d object with an unknown position and orientation. I want to determine this pose of the object. On the surface of the rigid object are 4 reference points. I know the spatial ...
0
votes
1answer
22 views

Locus problem solve it using simple mathematics

A line cuts $X$-axis at $A(7,0)$ and the $Y$- axis at $B(-5,0)$. A variable line $PQ$ is drawn perpendicular to $AB$ cutting $X$-axis at $P$ and $Y$-axis at $Q$. If $AQ$ and $BP$ intersect at $R$, ...
0
votes
1answer
21 views

What goes the Pappus' theorem says

I found the following statement: Let $A, C$ and $E$ be three distinct points on the line $l_1$ and $B,D,F$ three distinct points on the line $l_2$. Let us assume that $AB\cap DE=L$, $CD\cap FA=M$, ...
1
vote
1answer
12 views

Solving the euclidian distance squared to kernelize a Lagrangian dual

Homework question, looking for a hint on the following problem: I'm trying to solve this dual lagranging form (which could potentially be wrong already, but let's assume it is right) $\boldsymbol{x}$ ...
0
votes
0answers
10 views

Estimating the mean Euclidean distance between two overlapping, not-matching shapes

I’d like to determine the mean distance between two irregular overlapped, not-matching shapes ($X$ and $Y$). In $Figure 1$, $X$ is “visually above” $Y$, and that’s why we can’t see part of the $Y$ ...
1
vote
2answers
192 views

How to find the number of squares formed by given lattice points?

Let us say that we are N integer coordinates (x, y) - what would our approach be if we were supposed to find the number of squares we could make from those given n points? Additionally, if we were to ...
1
vote
1answer
37 views

Triangle orthocenter problem

I found a theorem written in a clumsy way. Is this theorem true? Let $ABC$ be a triangle and $DEF$ triangle made by the base points of altitudes of $ABC$. Then the center of an incircle of $DEF$ is ...
0
votes
0answers
23 views

Find the line that intercepts the lines $r$ and $s$ and forms congruent angles to the coordinate axes

I need to find the line $t$ such that it intercepts $$r:\frac{x-1}{3} = \frac{y-1}{2} = -\frac{z}{3}$$ and $$s:\\x=-1+5\lambda\\y = 1+3\lambda \\z = \lambda$$ And also, that forms congruente angles ...
1
vote
1answer
27 views

Find the line that passes by $P=(1,-2,3)$ and has angle $45$ and $60$ respectively with the $x$ and $y$ axis

I have $$P=(1,-2,3)$$ And the x-axis with direction vector $(1,0,0)$ and y-axis with direction vector $(0,1,0)$. By the angle formula: $$\cos\theta = \frac{|\vec u\cdot\vec v|}{||\vec u||||\vec ...
0
votes
1answer
31 views

How to prove the Pitot's theorem?

I read the following Pitot's theorem: A quadrilateral ABCD is tangential if and only if $AB+CD=AD+BC$, where $AB$ means the length of side $AB$. How can I prove it. I mean, the case $ABCD$ is ...
1
vote
2answers
27 views

The relation of angle between two slant faces of a pyramid and the angles between slant vectors

Have any of you seen this theorem of relationship of the angles between two slant faces of a pyramid and the angles between slant vectors, provided that two faces of corresponding to $\phi$ and $\eta$ ...
1
vote
1answer
52 views

Finding an angle between the side of a triangle and a segment from a point inside the triangle.

Question given below: ABC is a triangle and D is a point inside ABC such that: $$ m(\widehat{DCB})=m(\widehat{CBD})=18^{\circ}\\ m(\widehat{ACD})=24^{\circ}\\ m(\widehat{DBA})=12^{\circ}\\ ...
1
vote
1answer
18 views

Proof of a point beween two different points. (Geometry)

I'm struggling with some of the logic writing this proof. This is the question: Prove that if X is in AB (AB is a line segment) with X =/= B, then dist(AX) < dist(AB). Logically this makes ...
1
vote
1answer
25 views

Angles in a circle

I have troubles to prove the following: Let $\Gamma$ be a circle with center $O$, $a$ be a tangent to $\Gamma$, $A=a\cap \Gamma$, $D$ a point on $a$ and $B\in \Gamma$ such that $D$ and $B$ lies on ...
0
votes
1answer
21 views

Measure of angles is the same

I found the following theorem in a lecture notes without proof: Let $A, B, C, D, E$ and $F$ be points on the plane such that $\angle ABC$ and $\angle DEF$ are either both acute or they are both ...
1
vote
2answers
73 views

What are the ranges of triangle angles?

Lets say, that $\alpha \le \beta \le \gamma$. As shown here, $60 \le \gamma \lt 180$. What are the minimum and maximum values of $\alpha$ and $\beta$? The answer: $$0\lt \alpha \le 60 \\ 0 \lt ...
1
vote
1answer
33 views

What is the range of angle in front of longest triangle edge?

What is the minimum and the maximum values of the angle $\gamma$ in front of the longest triangle edge?
0
votes
1answer
25 views

How to prove the tangent secant theorem

I was reading the the following theorem: Let $A,B$ be two points on the circumference of a circle. Let $C$ be a point outside the circle. Then $\angle BAC=\frac{1}{2}\widehat{AB}$. Is there some ...
-4
votes
0answers
21 views

RMO and level of dificulty [duplicate]

What are the major topics in 1st stage 1 and how to prepare for major topics in the exam . Please give an quick answer.
0
votes
1answer
25 views

Geometric Construction

"Given in position the points $A,B$ and $P$ and a segment $m$, draw through $P$ a line $r$ in such a way that $A$ and $B$ be in opposite sides of $r$ and that the sum of the distances from $A$ and ...