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Questions tagged [euclidean-geometry]

For questions on geometry assuming Euclid's parallel postulate.

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concyclicity related to the Humpty point

$AD$ and $CE$ are the altitudes of triangle $ABC$, $M$ is the midpoint of $AC$. Circumcircles $\omega_1$ and $\omega_2$ of triangles $AEM$ and $CDM$, respectively, intersect at point $P$. $CP$ and $AP$...
Meison's user avatar
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1 answer
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Alternative proof of Apollonius's identity

How does one prove Apollonius's identity without coordinates or the law of cosines (or similar trigonometric laws)? Is there a known rearrangement proof similar to the many such proofs for the ...
Greg Nisbet's user avatar
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Conditions on a finite metric that guarantees embedding in Euclidean space? [duplicate]

If we have $n$ points in some metric space, do there exist coordinates for the $n$ points in an $n-1$ dimensional Euclidean space with exactly the same pairwise distances as in the original space? ...
user9998990's user avatar
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1 answer
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Tetrahedron analogue of a triangle Cevians property

It's a cute Olympaid geometry problem to prove that given a triangle $\triangle ABC$ with three Cevians $AA_1,BB_1, CC_1$ intersecting at an interior point $M$: $$ab+bc+ca+2abc = 1 \quad (1)$$ where $$...
dezdichado's user avatar
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Rational quantities associated with a bicentric heptagon

For odd reasons of my own, I would like to create a math puzzle involving a bicentric heptagon (i.e., a 7-sided polygon which has both an inscribed and a circumscribed circle). The puzzle is to be ...
tuna's user avatar
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0 votes
1 answer
37 views

How to express an angle between two angle bisectors in interior angles of a convex quadrilateral?

Given a convex quadrilateral $ABCD$, I would like to express the angle between angle bisector of internal and external angles of the opposite vertices in interior angles of $ABCD$. Here is the drawing ...
Rusurano's user avatar
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2 votes
3 answers
112 views

Find the segment "DC" in the obtuse triangle below

In an obtuse triangle $ABC$, obtuse at $B$, the internal bisector $AD$ is drawn and in $AC$ the point $"q"$ is taken such that $m∡𝐴⁢𝐷⁢𝑄=90^𝑜$. Calculate $DC$. If: $AQ = 10$ and $AB = BC$. $Answer:...
peta arantes's user avatar
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-3 votes
2 answers
64 views

If ABCD is a 2*2 square , E is midpoint of AB ,F is the midpoint of BC , AF and DE intersect at I , BD and AF intersect at H , Find area of BEIH. [closed]

This question is from 1991 AHSME problem 23 , In the solution of problem it is solved by using coordinate geometry . My question is can we solve using only Euclidian geometry + may be some ...
curious's user avatar
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Number of Tverberg Partitions [closed]

Tverberg's Theorem: A collection of $(d+1)(r-1) +1$ points in $\mathbb{R}^d$ can always be partitioned into $r$ parts whose convex hulls intersect. For example, $d=2$, $r=3$, 7 points: Let $p_1, p_2,...
D. S.'s user avatar
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1 answer
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Helly's theorem for $n\geq d+3$

Helly's theorem : Let $C_1,\ldots,C_n$, $n\geq d+1$, be convex sets in $\Bbb R^d$. Suppose every $d+1$ have a common intersection. Then they all have a common intersection. Proof: We're given that ...
D. S.'s user avatar
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3 votes
3 answers
125 views

Find the angle $x$ in the triangle $APQ $ below

iF $AB=AQ$ calcule $x^o$ (Answer: $30^o$) I try: $\triangle ABQ: \angle ABQ \cong \angle BQA = 90^o -2\alpha$ $\triangle ABC: \angle B = 180^o - 4\alpha -(60^o -2\alpha) =120^o -2\alpha $ $ \angle ...
peta arantes's user avatar
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+100

What is the number of facets of a $d$-dimensional cyclic polytope?

A face of a convex polytope $P$ is defined as $P$ itself, or a subset of $P$ of the form $P\cap h$, where $h$ is a hyperplane such that $P$ is fully contained in one of the closed half-spaces ...
A. H.'s user avatar
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1 answer
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Euclid's fourth postulate [closed]

I am currently reading through Roger Penrose's The Road To Reality. A question about Penrose's explanation for the necessity of Euclid's fourth postulate (Chapter 2, pg. 28-29 in pdf). Penrose writes: ...
Anton Everts's user avatar
1 vote
1 answer
118 views

Calculate the angle $x$ in the quadrilateral $ABCD$

Calculate the angle $x$ formed by the diagonals of a quadrilateral given $$\angle ABD = 50^o\\\angle EBC = 80^o \\ \angle BDA = 100^o \\ \angle BDC=30^o\\\angle CEB = x$$ (Answer:$60^o $) My try: $\...
peta arantes's user avatar
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0 answers
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Polygon Boundary in 3D Space

A simple polygon (interior included) is a manifold with boundary being homeomorphic to a closed disk. Its (manifold) boundary is a non-intersecting closed polygonal chain. Viewed as a subset of the ...
Wipetywipe's user avatar
1 vote
1 answer
37 views

Calculate the angle $x^o $ in the triangle below

Calculate the angle $x^o $(Answer:$25^o$) I try: $angle ABE = 180^o -65^o = 115^o $\ $\angle EBC = 180^o - 70^o - 45^o = 65^o \implies \angle ABE = 50^o$ $ \angle AEB = 180^o - 70^o =110^o $ $\angle ...
peta arantes's user avatar
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0 votes
1 answer
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Unit ball with dual set [closed]

Let $X\subseteq \mathbb{R}^d,$ we define the set dual to $X$, denoted by $X^*$, as follows: $$X^*=\{y\in \mathbb{R}^d \mid \langle y,x\rangle\leq 1, \forall x\in X\}.$$ Geometrically, $X^*$ is the ...
D. S.'s user avatar
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2 votes
2 answers
66 views

Finf the segment EC in the triangle ABC below

In a triangle $ABC$ where $AB = 6$ and $BC = 9$; the extension of the bisector of angle ABC intersects the (perpendicular) bisector of $AC$ at $P$ and is then drawn: PE⊥BC. Calculate $EC$. (Answer:1,5)...
peta arantes's user avatar
  • 7,031
1 vote
0 answers
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Set dual with half-spaces

Let $X\subseteq \mathbb{R}^d,$ we define the set dual to $X$, denoted by $X^*$, as follows: $$X^*=\{y\in \mathbb{R}^d \mid \langle y,x\rangle\leq 1, \forall x\in X\}.$$ Geometrically, $X^*$ is the ...
D. S.'s user avatar
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1 vote
3 answers
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Find the measure of $\measuredangle BAC$ in the triangle below given two sides and an angle

Em um triângulo $ABC$, se $∡𝐴⁢𝐶⁢𝐵=34^𝑜⁢30′$; $AB = 6$ e $BC = 10$. Calcular ∡𝐵⁢𝐴⁢𝐶. (Answer:$71^o 30'$) I try: T.coss: $6^2 = 10^2+AC^2-2.6.AC.cos \angle C \implies 12AC cos \angle C C$ $\...
peta arantes's user avatar
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0 votes
3 answers
53 views

How to prove opposite angle bisector theorem for convex quadrilaterals?

Let $ABCD$ be a convex quadrilateral with $BL$ and $DL$ be its angle bisectors. I want to know how to prove that the acute angle $\alpha$ between these bisectors is equal to $\frac{\left|\angle A - \...
Rusurano's user avatar
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3 votes
0 answers
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A beautiful property of two parabolas that intersect in four points

I have just come up with a very cool property of two parabolas intersecting at four points, I want to know whether this property is already known or not and how to prove it. We have two parabolas ...
زكريا حسناوي's user avatar
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0 answers
28 views

A detail in the proof of Killing-Hopf theorem for Euclidean surface.

I am reading the book Geometry of Surfaces by Stillwell. In chapter $2$, he proves the following theorem: Theorem: (Killing-Hopf) Each complete, connected Euclidean surface is of the form $\mathbb{R}^...
Zoudelong's user avatar
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5 votes
0 answers
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Existence of non-intersecting semicircular arcs connecting every points in a given set

This is a problem that I came up with a few months ago. Let $S$ be a set of $n$ points on a plane ($n \ge 3$), and let $e(c)$ denote the set of the two end points of the semicircle $c$. A ...
Kevin Cheng's user avatar
0 votes
2 answers
78 views

How to check if a point lies inside a tetrahedron [closed]

Given a tetrahedron $T$ with apex $O$ and vertices $V_1, V_2, V_3$ and point $P=(x_1, y_1, z_1)$ in $3$D space. I need to check programmatically if $P$ lies inside $T$. We can also assume the angle ...
M a m a D's user avatar
  • 451
-1 votes
1 answer
59 views

Perpendiculars in 3 dimensions [closed]

I was looking for the proof for the distributive property of the dot-product with the definition that $\vec a \cdot \vec b = ab\cos\theta$ as I came across the second proof of this page: https://...
Artin8476's user avatar
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0 answers
63 views

Prove centre is inside cyclic quadrilateral with perpendicular diagonals

Let $ABCD$ be a convex cyclic quadrilateral, and the diagonals $AC$ and $BD$ are perpendicular. The circumcircle of $ABCD$ has centre $O$. I am trying to prove that the centre $O$ is inside $ABCD$. I ...
wenbang's user avatar
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0 answers
50 views

How did Euclid's proof of the isosceles triangle theorem (*pons asinorum*) differ from the High School geometry proof?

Consider the first theorem proved in a High School geometry class. Theorem. The opposite angles of an isosceles triangle are congruent. Proof. Construct an angle bisector through the vertex of the ...
Fomalhaut's user avatar
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5 votes
1 answer
672 views

The center of gravity of a triangle on a parabola lies on the axis of symmetry

About an hour ago, I discovered a beautiful property of a parabola. If a circle intersects a parabola at four points, one of which is the vertex of the parabola, then the center of the triangle, ...
زكريا حسناوي's user avatar
2 votes
4 answers
106 views

Is there is a formula to calculate the coordinates of the orthocenter of a triangle?

I'm trying to find the coordinates of the orthocenter (the intersection point of all altitudes) of a triangle given its vertices' coordinates $A=(x_1, y_1), \ B=(x_2, y_2) , \ C=(x_3, y_3)$. I ...
pie's user avatar
  • 6,620
4 votes
1 answer
135 views

Simplicial Generalization of Pythagoras

I recently heard about a claim that For a triangle in 3-space, its area squared equals the sum of squares of areas of its projections onto three pairwise orthogonal planes. I currently don't have ...
Dr. Richard Klitzing's user avatar
2 votes
1 answer
55 views

Finding the circumradius of a cyclic hexagon, given three non-consecutive sides and the fact that the midpoints of all sides are also cyclic

Cruel Geometry Question, Mock AIME-i 2015: Let $A_1A_2A_3A_4A_5A_6$ be a hexagon inscribed inside a circle of radius $r$. Furthermore, for each positive integer $1 \leq i \leq 6$ let $M_i$ be the ...
CLASH ROYAL's user avatar
0 votes
1 answer
103 views

A counterexample that $C^\infty(\mathbb{R}^n)$ is not complete.

Can anyone give an example which can directly indicate that the following norm space is not complete: $$ C^{\infty}(\mathbb{R}^n)\cap W^{1,p}(\mathbb{R}^n) = \{f\in C^{\infty}(\mathbb{R}^n): f \in L^p(...
tianJ's user avatar
  • 41
3 votes
2 answers
90 views

Find $\angle BCD$

A convex quadrilateral $ABCD$ satisfies $\angle CAB= \angle ADB=30^\circ,\angle ABD=77^\circ$ and $ BC=CD.$ Find $\angle BCD$ My attempt: Constructed the perpendicular bisector of $DB$ to meet $AD$ ...
Bijesh K.S's user avatar
  • 2,646
8 votes
1 answer
134 views

Double Contact Chained Ellipses Problem

A few years ago, when I played around with GeoGebra, I came up with the following conjecture. Conjecture Let $n\in\mathbb{N}, n\geq3$. Let $E$ be an ellipse, and let $E_{1}, E_{2}, \dots, E_{n}$ be ...
K. Miyamoto's user avatar
1 vote
3 answers
43 views

Properties that relate to the chord of a parabola passing from the perpendicular projection of the focus point on the parabola guide [closed]

Now I remembered my previous question about finding the harmonic mean using a parabola and my answer, which included a second method. That second method inspired me to try more in this configuration ...
زكريا حسناوي's user avatar
2 votes
1 answer
73 views

Determine the angle $\angle DEC$ in a triangle (Euclidean Geometry)

Any ideas how to find the angle $\angle DEC$ in the following situation shown in the image: In the above figure we have that $\angle BAC = 90, \angle ABD = \alpha, \angle DBC = 2\alpha$, and $\angle ...
ChrisNick92's user avatar
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2 votes
3 answers
130 views

Determine the distance from a point $P$ to the center of the circle as a function of the radius $R$

Let AB be the diameter of a circle with center O and radius R. On the extension of AB we choose a point P (PB<PA). Starting from P we take a secant that cuts the circle at points M and N (PM<PN),...
peta arantes's user avatar
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0 votes
1 answer
33 views

reflexive property of congruence

While exploring basic geometry with my artist friend, she raised a question that neither of us could answer satisfactorily. Until today I thought that Hilbert's axiomatization of Euclidean geometry is ...
Ehsan Amini's user avatar
0 votes
1 answer
21 views

A regular plane curve with no segment parameterisable as a line segment has a constant binormal? A clothoid?

This is an exercise in Valter Moretti's Analytical Mechanics Exercises 2.21 (1) Consider the curve $\Gamma,$ of class $C^{1}$ and regular in $\mathbb{E}^{3},$ parametrised by its arclength $P=P\left(...
Steven Thomas Hatton's user avatar
0 votes
1 answer
52 views

Find the side size of the square in the figure below

In the plane figure below, we have a parallelogram $ABCD$ and three squares, two of which have side measurements equal to the sides of the parallelogram and the largest of which has side measurements ...
peta arantes's user avatar
  • 7,031
1 vote
2 answers
41 views

Alternate Proof for Sum of Sides of a Triangle Inequality

I recently stumbled upon an idea for a proof for the sum of two sides of a triangle inequality. Note that I am just a high school student and feel free to correct me wherever if I am wrong. Statement/...
Rishwanth's user avatar
1 vote
1 answer
270 views

Sum of the vectors from centre $O$ to the polygon vertices

I'm attempting to calculate the sum of the vectors from the center of a regular polygon to each of the vertices. I have already solve it in a complex analysis manner: To represent the vertices of a ...
Hank Wang's user avatar
4 votes
2 answers
107 views

What's the maximum of $(\sin A-\sin B)(\sin B-\sin C)(\sin A-\sin C)$ in one triangle?

What's the maximum of $(\sin A-\sin B)(\sin B-\sin C)(\sin A-\sin C)$ in one triangle (suppose $\sin A\geq\sin B\geq\sin C$)? The result from Mathematica is that when $A\approx1.95398,B\approx0.913057,...
grj040803's user avatar
  • 701
0 votes
1 answer
50 views

Understanding a hint from Coxeter

The following problem posed in Coxeter and Greitzer's Geometry Revisited is readily proved by angle chasing (stick angles at $A,B$, chase). I am seeking the proof indicated by the author's hint on ...
RobinSparrow's user avatar
  • 2,042
2 votes
2 answers
112 views

Sum of powers of distances between points in concentric circles

Let we have two concentric circles. Inner circle contains one point and outer circle - $N$ points as regular polygon vertices. It is well known that the sum of the squares of the distances from any ...
lesobrod's user avatar
  • 804
1 vote
1 answer
41 views

Calculate the product of the segments described below

Points $A0, A1, A2.....A_{2n}$ divide a circle of radius whose size is $R$ into an odd number of congruent parts, $B$ is a point diametrically opposite to point $A0$. Calculate: $BA1. BA2. BA3. BA4. .....
peta arantes's user avatar
  • 7,031
2 votes
1 answer
89 views

Find the size of the radius of the nth circle.

In the figure shown, calculate the size of the radius of the nth circle. (Answer:$\frac{R}{n^2+2}$) I calculated the radius of the first two circles. How to extrapolate this resolution to other rays? ...
peta arantes's user avatar
  • 7,031
0 votes
0 answers
36 views

Triangular inequality on the angles of the faces of a trihedron [duplicate]

I just saw this theorem which I'm trying to prove but got stuck. Given a trihedron like the one below, whose angles between its edges are $\widehat{bc}, \widehat{ac}$ and $\widehat{ab}$, then prove ...
hellofriends's user avatar
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1 vote
0 answers
37 views

Proof of Thomson cubic pivotal property without coordinates

The Thomson cubic is defined as the cubic going through A,B,C, the three side midpoints, the three excenters. Is there a way to prove its pivotal property (any two isogonal conjugates on it have a ...
user118161's user avatar

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