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Questions tagged [euclidean-geometry]

Geometry assuming the parallel postulate: 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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Prove midpoint is fixed

Suppose we have a fixed triangle $ABC$. $M$ is the midpoint of $BC$ and $D$ is a variable point on segment $BC$. $D'$ is the reflection of $D$ across $M$. The circumcircles of $\triangle ADM$ and $\...
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41 views

How many perpendiculars can be dropped from a given point to a given conic?

Could you give me any hints, please: how many perpendiculars can be dropped from a given point to a given conic?
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19 views

Why $\phi$ is a homography?

Let $C$ be a smooth conic on euclidean plane. Let $p$ be a point of $C$. Let's choose an arbitrary point $a$ of $C$. Let's define $\phi:$ $C \rightarrow C$, $b=\phi (a)$, $b$ is a point of conic such ...
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4answers
63 views

Proving that, for an acute $\triangle ABC$, $\sin A + \sin B+\sin C\gt \cos A+\cos B+\cos C$

I need to prove or disprove that in any acute $\triangle ABC$, the following property holds: $$\sin A + \sin B + \sin C \gt \cos A + \cos B + \cos C$$ To begin, I proved a lemma: Lemma. An ...
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1answer
44 views

How to find the value of $\alpha$ using the attached figure?

I tried to crack this task firstly using constructions like parms and equilateral triangles. This failed since i obtain zero. Also i think in using cosine and sine law but without any important result....
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1answer
22 views

Trapezium: length of a line of segment

Trapezium In the figure PQRS is a trapezium with PQ parallel to SR. The diagonal of the trapezium meet at X. U lies on SP and T lies on RQ such that UT is a line segment through X parallel to PQ. ...
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2answers
45 views

Trigonometry puzzling problem

Contructed the figure here reported and known that $AB=2r$, $AC=r\sqrt{2}$ and fixed $x:=P\hat{A}C$, it is asked to find $$ y = CK + PH \sqrt{2} + PK $$ as a function of $x$. I started with $$PH =...
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Find the Measures of the Missing Angles

Find the measures of the angles $x$ and $y$ in the diagram above. I've tried using angles in the triangles and quadrilateral, exterior angles, and parallel lines. Everything I come up with reduces ...
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2answers
28 views

Geometry question from Triangles

Suppose $AD$ bisects angle $A$ of triangle $ABC$ and meets $BC$ at $D$, and let $S$ and $S'$ be the circumcenters of triangles $ABD$ and $ACD$ respectively. Show that $$\frac{SD}{S'D}=\frac{BD}{DC}.$...
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53 views

Question of Triangle of Geometry

In triangle ABC,we have $AB>AC$ . If A' is the mid point of BC,AD is the altitude through A and if the internal and external bisectors of angle A meet at BC at X and X' respectively ,prove that $A'...
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Triangle inequality for angles in Euclidean space [duplicate]

Is there any simple proof of the following statement: for all vectors $ v,w,u\in V\setminus\{0\} $, where $ V $ is a Euclidean space, inequality $$ \angle(u,v)\le\angle(u,w)+\angle(w,v)$$ holds. ...
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Write down the eqn. of the line that passes through the points $(4, -2, 1)$ and $(6, 0, 3)$ in all three forms.

I'm stuck because I can't find the parallel vector. Do you know how to find it? I'd just assumed that vector r0 = <4, -2, 1>.
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1answer
17 views

Solving plane geometry using vectors and conditions on angles

I can use simple vectors to work out Euclidean properties of plane figures such as the location of the intersection of the bisectors of the sides of triangles (the centroid). Is it possible to use ...
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0answers
19 views

Definition of positively oriented basis

Let $(U_\mathbb{R},A)$ be an oriented $m$-dimensional Euclidean space $(U_\mathbb{R},A)$, where $A$ is an alternating unimodular $m$-th order tensor, and subset $B=\{\vec{u}_1,\cdots,\vec{u}_m\}$ a ...
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1answer
52 views

radius of the inscribed sphere of the pyramid in $\mathbb{R^5}$

How to find the radius of the inscribed sphere of the pyramid in $\mathbb{R^5}$ with the vertex in $(1,0,0,0,0)$, which base is a regular $4$-dimensional simplex, luying in the hyperplane $x_1=0$ with ...
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3answers
69 views

Straight line involving polar form

A variable line $L$ through $P(3,4)$ meets lines $x-y+6=0$ and $x-y+10=0$ at $A$ and $B$. A point $Q$ is on $L$ such that $PQ^2=PA.PB$ . Find locus of $Q$. Attempt: If variable line $L$ makes angle $...
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2answers
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Prove that $NP = NQ$.

$M$ is the orthocenter of $\triangle ABC$. $D$ is a point outside of $\triangle ABC$ such that $\widehat{ABD} = \widehat{DBC} = \widehat{DMC}$. $N$ is the orthocenter of $\triangle BCD$. Let $P$ and $...
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Is there a point $H$ such that $\frac{AH \cdot DM}{HD^2} = \frac{BH \cdot EN}{HE^2} = \frac{CH \cdot FP}{HF^2}$?

$H$ is a point in non-isoceles triangle $\triangle ABC$. The intersections of $AH$ and $BC$, $BH$ and $CA$, $CH$ and $AB$ are respectively $D$, $E$, $F$. $AD$, $BE$ and $CF$ cuts $(A, B, C)$ ...
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1answer
77 views

Shortest distance between two rectangles in 2D

Given the center points and dimensions of two rectangles and the angles $\theta_1$ and $\theta_2$, how to calculate the shortest distance between two rectangles ($d$)? $\theta$ is the angle that the ...
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1answer
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If two medians are congruent… is the triangle isosceles in a Hilbert plane?

If $ABC$ is a triangle for which two medians are congruent... is it true that the triangle $ABC$ is isosceles in a general Hilbert plane? I am having a little bit of trouble trying to prove this, if ...
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1answer
32 views

Help with proving the following points are collinear

Let BC be the shortest side of $\triangle$ABC. Let P be a point in AB such that $\angle$PCB=$\angle$BAC and Q be a point in AC such that $\angle$QBC=$\angle$BAC. Prove that the line that passes ...
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44 views

What Type of Curve Is This?

What curve is the longest distance between points a and b such that (i) one gets closer to point b at all times; (ii) it can be graphed as a function; and (iii) it is symmetrical? I would think each ...
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Geometric locus of $P$ [closed]

Let $\angle XOY$ and $A, B, C\in (OX$ s.t. $OA<OB<OC$ and $M\in OY$ a mobile point.The line bisector of $\angle XOY$ intersects $CM$ in $N$ and $AN\cap BM =\lbrace P \rbrace$. Find the geometric ...
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1answer
43 views

Is this a valid proof of the parallel postulate?

Let x1 be a line with a slope of tan(n) where 0 ≤ n < π/2. x1 = tan((π/2)-n)y, 0 < n ≤ π/2 Let y2 be a line with a slope of tan(m) where ...
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1answer
45 views

a circle and a parabola have 3 intersection points

Is it possible that a circle and a parabola on a euclidean plane have 3 intersection points and the center of the circle does not lie on the axis of parabola?
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1answer
84 views

In a triangle $ABC$, prove that $\angle AST\equiv\angle OSB$ where $T$ is the midpoint of AH and $S=BC \cap L$ (L = tangent line in A to circumcircle) [closed]

Let $ABC$ a triangle and $H$ and $O$ its orthocenter and circumcenter. Denote by $T$ the midpoint of the segment $AH$ and by $S$ the intersection of the tangent in $A$ to the circle $ABC$ and the line ...
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5answers
215 views

If $\sin(18^\circ)=\frac{a + \sqrt{b}}{c}$, then what is $a+b+c$? [duplicate]

If $\sin(18)=\frac{a + \sqrt{b}}{c}$ in the simplest form, then what is $a+b+c$? $$ $$ Attempt: $\sin(18)$ in a right triangle with sides $x$ (in front of corner with angle $18$ degrees), $y$, and ...
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1answer
46 views

Show that a triangle is equilateral

A circle crosses the sides of a triangle, dividing each of them into three equal parts. Prove that the triangle is equilateral. I think that the best way is to show that $\angle BAC = \angle ABC$, ...
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1answer
63 views

Look for a clever geometric approach to solve a problem with area.

In the acute $\triangle ABC$, the position of each point is as shown, $F$ is the midpoint of $BD$, $G$ is the midpoint of $CE$, $F$ is not on $CE$, and $G$ is not on $BD$. It is known that $S_{\...
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1answer
127 views
+50

Find which conditions must parameters $a$ and $b$ meet so there's exist an orthonormal basis

In $\mathbb{E^3}$ we have the plane $\pi:x-y+z-3=0$, the line $r:(2,0,1)+t(1,1,0),\ t\in\mathbb{R}$, and the point $P=(3,0,3)$. Which conditions must parameters $a$ and $b$ meet so there's exist an ...
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1answer
32 views

Euclidean Geometry versus Analytic Geometry versus Affine Geometry?

What are the relationships (connections) among: Euclidean (or Plane) geometry Analytic geometry Affine geometry How do these things relate? I know that this is a very general question, so I'm ...
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3answers
56 views

Prove angle addition holds for $\mathbb{R}^n$

Define $\theta(u,v)=\cos^{-1}(\frac{u\cdot v}{|u||v|})$ be the angle between $u,v\in \mathbb{R}^n$, where $u\cdot v$ is the standard inner product and $|x|=\sqrt{x\cdot x}$ for all $x\in \mathbb{R}^n$....
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2answers
79 views

Existence of $1$-Lipschitz map between triangles

Consider two (Euclidean) triangles $T$ and $T'$. Let's say that $T$ majorizes $T'$ if there exists a 1-Lipschitz map that sends vertices to vertices and sides to sides (for some labeling of the ...
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1answer
44 views

$|\partial \Lambda_n| = 2d(2n - 1)^{d-1}$

In the lattice $\mathbb{Z}^d$, consider the norm $\vert\vert \cdot \vert \vert \colon \mathbb{Z}^d \rightarrow [0,+\infty)$, given by $$\vert \vert \lambda \vert\vert:= \displaystyle\max_{1 \leq i \...
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1answer
13 views

Finding algebraic expression of a parallepiped given directions and side lengths

Let $u_1,\dots,u_n$ be linearly independent unit vectors in $\mathbb R^n$. Let $T$ be the parallepiped centred at $0$, with sides parallel to $u_i$ and side lengths $l_i$, $i=1,2,\dots,n$. My question ...
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2answers
30 views

A circle inscribed in a trapezoid; $\angle BCH$

$ABCD: AB ||CD, AB>CD, AD=BC$ $k(O)$ inscribed $DH \bot AB,H \in AB$ and $\angle ADC = \gamma$ $\angle BHC, \angle BCH =$ ? I have tried to show that $\triangle BCH$ is isosceles, but when I ...
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0answers
40 views

A circle inscribed in a quadrilateral

A circle is inscribed inside a trapezoid $ABCD$ ($AB || CD$). $M, N, P, Q$ are the midpoints of the sides $AB, BC, CD, AD$, respectively. If $AD = d$ and $BC=c$, express the perimeter of $MNPQ$. (...
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2answers
63 views

Prove that $\frac{1}{AM \cdot BN} + \frac{1}{BN \cdot CP} + \frac{1}{CP \cdot AM} \le \frac{4}{3(R - OI)}$.

$(O, R)$ is the circumscribed circle of $\triangle ABC$. $I \in \triangle ABC$. $AI$, $BI$ and $CI$ intersects $AB$, $BC$ and $CA$ respectively at $M$, $N$ and $P$. Prove that $$\large \frac{1}{AM \...
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1answer
14 views

Nine point Circle right angled and tangent

I have been working on the following tasks for some time now and I do not know how to solve it. $(A, B, C)$ is a triangle and $K$ is his nine point circle. Show it: a) $(A, B, C)$ is right-angled if ...
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1answer
18 views

Orthogonal complement to a graph of a linear map

Let $E$ and $F$ be two Euclidean/Hermitian vector spaces and $f:E\rightarrow F$ a linear map. Let $\mathcal{G}(f)<E\oplus F$ be the graph of $f$. Assume if it helps that $\{e_i\}_i$ is an ...
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2answers
76 views

Find chord length given diameter and two other chords

Problem: I'm asked to find the length of $m$, given the following diagram. Note that $\overline{AC}$ = 1 and $\overline{CD} = 1$ and that $\overline{AB}$ is a diameter whose length is 4. Attempt: ...
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1answer
12 views

Dihedral angle of a regular simplex in $n$ dimensions

For the regular simplex on $(n+1)$ points in $n$ dimensions, what is the dihedral angle i.e. the angle between two of the faces?
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22 views

Partial derivative of euclidean distance

Hello dear mathematicians, I have two points in R3 between which I calculated the euclidean distance. Now I want to find the partial derivative with respect to the first and the partial derivative ...
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0answers
44 views

A geometric problem for a quadrilateral

1) I have to calculate the area of a kite $$ABCD$$ with $$AB-CD=(\sqrt {2}+1)(\sqrt{3}+1)$$ and $$11 \angle A= \angle C.$$ 2) A second question is that if $$2 AB^2+AC^2+2 AD^2=4 BD^2$$ then there is ...
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1answer
34 views

Simple problem in Euclidean Geometry — Find the radius of a circle

A student of mine brought the following question to my attention. I am currently not able to solve it, any help would be appreciated. It should be a simple circle theorem that I have now forgotten. ...
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3answers
75 views

Prove that a line touches a circle [closed]

Let $I$ be the incenter of $\triangle ABC$. The circle passing through $I$ and centered at $A$ meets the circumcircle of $\triangle ABC$ at points $M$ and $N$. Prove that the line $MN$ touches the ...
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1answer
24 views

Show that MN and MP are angle bisectors

A $\triangle ABC$ is drawn ($\angle C = 90^\circ$), in which $CL$ $(L \in AB)$ is bisector. The circle $k$ with diameter $CL$ intersects AB, BC and CA, respectively, in $M$, $N$ and $P$. Show that $...
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1answer
62 views

How to find the value of x?

I was trying to find the value of X for an hour now but don't know how to start and I don't know why A= 42.9cm2. I mean wasn't A supposed to be 16.5cm2. Thank you in advance.
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4answers
95 views

area of quadrilateral! only areas given!

Find the $$ar(CEF)+ar(FGB) =\;\;?$$ I am really stuck on this ... spend some hours... did not what to solve and how to proceed? Any suggestions ? Hints also work :)
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0answers
22 views

What condition of point $M$ and $P$ needs to be added such that $PK$ passes through the midpoint of $AG$?

$D$ is the midpoint of the smaller arc of $BC$ of the circumscribed circle of $\triangle ABC$ ($AB < AC$). $K$ is a point on $AC$ such that $DK = DC$. $AD \cap BK = E$, $EF \parallel BC$ ($F \in AC$...