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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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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12 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
28 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
30 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
36 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 $v$, ...
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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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29 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 ...
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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
23 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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18 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
31 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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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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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
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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1answer
62 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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1answer
39 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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36 views

Geometry using centers of mass [closed]

Show using cneters of mass that the angle bisectors of triangle $\triangle ABC$ are concurrent at a point $I$ where $I$ lies on bisector $AP$ at a position such that $\frac{AI}{AP} = ...
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2answers
157 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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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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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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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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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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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
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
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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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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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
32 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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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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0answers
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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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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 ...
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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
47 views

Find the maximum possible area for the triangle

Two vertices of an isosceles triangle are (1,2) and (4,6). The inradius of the triangle is $\frac{3}{2}$. Find the maximum possible area for the triangle. My work, for the two possible structures of ...
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1answer
34 views

slope of a curve in $\mathbb{R}^3$

The surface given by $z = x^2 -y^2$ is cut by the plane given by $y = 3x$, producing a curve in the plane. Find the slope of this curve at the point $(1, 3, -8)$. My answer is: $$f(x, y, z) = x^2 - ...
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1answer
26 views

Line goes through the center of circle [closed]

How can I prove this theorem which looks easy. Circle $C$ has a center $O$. Take four points $A,B,D,E$ on the circumference of $C$ such that $AB$ is parallel to $DE$. Let $F$ and $G$ be midpoints of ...
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1answer
27 views

Prove that $\angle BAC + \angle OAP = 180^\circ$

Prove that if you construct two circle centered at O and P and intersecting at A with tangent lines BA and CA. Prove that $\angle BAC + \angle OAP = 180^\circ$. I'm having trouble just starting the ...
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30 views

Derivation of Steepest Descent Direction used in Line Search Methods

In the numerical optimization text I am reading, the Steepest Descent Direction was derived by considering $$ \min_{||p||_2\leq 1} p^T\nabla f(x_k) $$ This resulted in $$ p_k=-\frac{\nabla ...
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21 views

Synthetic proof of curvature formula

The radius of curvature of a curve $\gamma:I \to \mathbf{R}$ parametric by arc length is $||\ddot \gamma||^{-1}$. I want to demonstrate this using synthetic geometry. Let $A$, $B$ and $C$ be three ...
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15 views

How To Find a Set of Points Farthest Apart Within 3D Solid

I am trying to find out a method to solve the following problem: There are two parameters: 1) There is a solid 3D region plotted in a cartesian coordinate system. 2) There is a number of points that ...
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4answers
296 views

How to show that these two lines are perpendicular?

Let $AEE'$ be an isoceles triangle with $\angle EAE'=90^\circ$ such that $AE=AE'$ and such that $A$, $E$ and $E'$ lie on the circle $c_1$. Let $ADD'$ be an isoceles triangle with $\angle ...
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1answer
46 views

Chords joining midpoints of four arcs on a circle

Let $W$, $X$, $Y$, $Z$ be the midpoints of arcs $\stackrel{\frown}{AB}$, $\stackrel{\frown}{BC}$, $\stackrel{\frown}{CD}$, $\stackrel{\frown}{DA}$, respectively. Show that the chords $WY$ and $XZ$ ...