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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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 $v$, ...
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1answer
56 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 ...
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48 views
+50

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 ...
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2answers
19 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 $ABCD$ ...
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0answers
19 views

Existence of half-planes with respect to regular open sets of the Euclidean plane

Let $\langle\mathrm{r}\mathscr{O},\mathord{\subseteq}\rangle$ be the complete Boolean algebra of open domains (regular open sets, these that are equal to the interior of their closure: ...
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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 ...
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1answer
31 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.
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1answer
22 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 ...
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2answers
268 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 ...
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1answer
597 views

Intersection of Two Circles

I have two circles as: $C_1: (x-x_1)^2+(y-y_1)^2=r_1^2$ and $C_2: (x-x_2)^2+(y-y_2)^2 =r_2^2$ and these circles have non-empty intersection. In other words $\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\leq ...
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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?
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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?
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1answer
26 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
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0answers
19 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 ...
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1answer
801 views

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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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 ...
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2answers
338 views

Trilateration with unknown fixed points

I am able to measure my distance to a set of (about 6 or 7) fixed but unknown points from many positions. The difference in position between measurements is also unknown. I believe that I should be ...
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1answer
22 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 ...
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1answer
13 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 ...
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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 ...
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1answer
40 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 ...
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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 ...
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2answers
191 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 ...
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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 ...
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1answer
642 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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1answer
46 views

Triangle inequality for angles

For points $O,A,B,C$ in $\mathbb{R}^{3}$, I was trying to show $\angle AOC \le \angle AOB +\angle BOC$. I could show this when all angles were acute. First, I set $O$ to be the origin and $A,C$ to ...
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0answers
24 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 ...
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1answer
20 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 $R$, ...
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1answer
20 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 $AB\cap DE=L$, $CD\cap FA=M$, ...
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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) $\boldsymbol{x}$ ...
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8 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 $Y$ ...
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3answers
174 views

Can a tetrahedron lying completely inside another tetrahedron have a larger sum of edge lengths?

Find 2 tetrahedrons $ABCD$ and $EFGH$ such that $EFGH$ lies completely inside $ABCD$. The sum of edge lengths of $EFGH$ is strictly greater than the sum of edge lengths of $ABCD$. I am completely ...
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1answer
36 views

Power of a point problem

There was a Finnish matriculation examination there was the following question: Consider a circle and a point $P$ outside the circle. From the point $P$ draw two lines such that each of the line ...
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1answer
24 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 ...
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1answer
36 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 ...
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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 ...
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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 ...
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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 $\eta$ ...
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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 ...
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285 views

Hidden geometrical gems in Euclid's Elements?

I am teaching a course in Euclidean geometry at the University of South Carolina, and it seemed highly appropriate and interesting to read Euclid himself. (See here for a wonderful, and completely ...
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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}\\ ...
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1answer
326 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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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 ...
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1answer
24 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 ...
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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 ...
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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 ...
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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?
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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.
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228 views

What is the smallest possible angle of this polygon?

A convex polygon contains a square with side-length 1 and is contained in a parallel square with side-length 2 (which is its smallest containing square). What is the smallest possible angle of the ...
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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 ...