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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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1answer
28 views

I have to show that $P, R, Q, S $ are on a circle.

Let $ABCD $ a paralelogram and $H $ the hortocenter of $\triangle ABC $. Let $PQ $ , $RS $ trough $H $ s.t. $PQ|| AB $ and $RS||BC $ and $P\in [DA], R\in [AB], Q\in [BC] , S\in [CD] $. I have to ...
2
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2answers
27 views

A conjecture about the sum of the areas of $3$ triangles built on the sides of any triangle (by means of centroid/orthocenter)

Given any triangle $\triangle ABC$, let us draw its orthocenter $D$. By means of this point, we can draw three circles with centers in $A,B,C$ and passing through $D$. These circles intersect in the ...
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1answer
20 views

Prove that in $\triangle ABC$ ,$QP\parallel BC$

In the triangle $\triangle ABC$ , the point $M$ is between $B$ and $C$. And also the lines $MP$ and $MQ$ are bisectors of $\angle AMC$ and $\angle AMB$. It means that: $$\angle AMP=\angle PMC$$ $$\...
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1answer
13 views

How to define two dimensional plane?

I know that we use three non-linear (at least one of them not linear) dots to define a 3d plane, but how do we define a 2 dimensional plane ? I am very bad at math and yes I have searched that in the ...
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12 views

special interior angles of regular $n$-gon

Let $K$ be a regular $n$-gon in the plane. Assume the following: $$n\in\Bbb N,\quad n\geq3\\I=\{0,1,...,n-1\}\\ \forall i\in I, \quad M(i):=\operatorname{mod}(i,n)$$ And we define $P_i$ as the $i$-th ...
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11 views

Three points fixed by the composition of an two isometries

I am in the final step of a proof on classifying the symmetries of $\mathbb{R}^2$. Suppose we have some symmetry $\sigma$ that fixes at least two points, say $A$ and $B$. Then consider $C$ which ...
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1answer
13 views

Any symmetry that fixes three non-collinear points is the identity

I am asked to finish the following sentence: Let $\sigma$ be an isometry on $\mathbb{R}^2$, suppose it fixes the points $A$ and $B$ Suppose $\sigma$ also fixes a third point $C$ which is not on the ...
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29 views

Is there a name for the minimal surface connecting two straight line segments in 3-dim Euclidean space?

In the $2$-dim Euclidean plane, the minimal curve that joins (and necessarily contains) two given points is a straight line. Here what is being minimized by the curve is the $1$-dim measure of the $1$...
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2answers
30 views

Prove that here $BE×DF$ is fixed

We have a parallelogram namely $ABCD$. Then we draw a line from the vertex $A$ to: 1- Cross a point (namely $E$)on the side $BC$. 2- Cross a point (namely $F$) along the side DC. Now we must prove ...
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1answer
19 views

Proove Line Circumscribed Circle; Incircle; Excircle

How can I proove, that the circumscribed circle of a triangle does exactly cross the middle of the line that goes from the incenter of the incircle of the triangle to the excenter of the excircle of ...
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2answers
72 views

Finding the side length of an equilateral triangle having three inscribed $120^\circ$ sectors in a certain arrangement

How do i even start this question? I thought of using length of tangent equal from exterior point but still of no use.
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1answer
23 views

Finding Angle in Triangle inside square

I need help to solve the following problem. Consider a square in the Euclidean plane with vertices $A,B,C,D\in\mathbb{R}^2$ (named in counterclockwise direction). Moreover, let $P$ be a point ...
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0answers
28 views

On Geometry problem solving methods

Till now, among all types of Geometry problem solving methods I have found these:. 1. Euclidean geometry (including trigonometry). 2. Analytical geometry (including all kinds of coordinates, ...
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2answers
33 views

Elementary euclidean geometry problem

The problem states: Consider a triangle $\Delta{ABC}$ in which $AC\gt AB$. A half line with origin in B cuts AC in D such that the angles $\angle ABD$ and $\angle BCD$ are equal. Prove that $(...
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3answers
206 views

Angles of triangle $\triangle XYZ$ do not depend on the position of point $P$ (proof needed)

Let $ABCD$ be a fixed convex quadrilateral and $P$ be an arbitrary point. Let $S,T,U,V,K,L$ be the projections of $P$ on $AB,CD,AD,BC,AC,BD$ respectively. Let $X,Y,Z$ be the midpoints of $ST,UV,KL$. ...
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1answer
70 views

A $10$-point conic through midpoints and intersections of a quadrilateral's sides and diagonals, and the point of concurrence of three related circles

I was messing around on GeoGebra when I discovered a mysterious hyperbola in a quadrilateral, among other really fascinating theorems. I'm not $100\%$ certain what I have found is true, as I have no ...
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4answers
894 views

Proof without words of a simple conjecture about any triangle

Given the midpoint (or centroid) $D$ of any triangle $\triangle ABC$, we build three squares on the three segments connecting $D$ with the three vertices. Then, we consider the centers $K,L,M$ of the ...
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3answers
78 views

$M$ is a point in an equaliateral $ABC$ of area $S$. $S'$ is the area of the triangle with sides $MA,MB,MC$. Prove that $S'\leq \frac{1}{3}S$. [on hold]

$M$ is a point in an equilateral triangle $ABC$ with the area $S$. Prove that $MA, MB, MC$ are the lengths of three sides of a triangle which has area $$S'\leq \frac{1}{3}S$$
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0answers
43 views

Let $M$ be an internal point of a triangle $ABC$ with $BC=a$, $CA=b$, and $AB=c$. Show that $\frac{MA}{a}+\frac{MB}{b}+\frac{MC}{c}\geq \sqrt{3}$. [on hold]

Let $M$ be an internal point of a triangle $ABC$ with $BC=a$, $CA=b$, and $AB=c$. Prove that $$\frac{MA}{a}+\frac{MB}{b}+\frac{MC}{c}\geq \sqrt{3}.$$
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1answer
79 views

Show that $BP+BQ=2PQ $

Let consider a circle of diameter $CA $ and $B\in CA $ such that $A\in [CB] $ and $AB=\frac {CA}{2} $. If $M \in [CA] $ such that $AM=\frac {CA}{3} $ and $P, Q $ on circle such that $P, M, Q $ ...
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2answers
64 views

Prove that $|A_1A_2|^2+|A_2A_3|^2+\ldots+|A_{n-1}A_n|^2+|A_nA_1|^2\leq 9R^2$. [on hold]

A polygon $A_{1}A_{2}...A_{n}$ has a circumscribed circle with radius $R$. Prove $$|A_1A_2|^2+|A_2A_3|^2+\ldots+|A_{n-1}A_n|^2+|A_nA_1|^2\leq 9R^2.$$
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1answer
47 views

Prove that $NK$ is tangent to the circumcircle of $\Delta KEF$ .

Consider a circle $O$ with radius $R$. $ABCD$ is cyclic quadrilateral, the intersection of $AC$ and $BD$ is $K$. $P$ and $Q$ are respectively the midpoints of $KD$ and $KC$. The intersection of $AP$ ...
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1answer
11 views

Prove that there exist exactly two motions which map two line segments of the same length to each other in a Euclidean Frame

Given two line segments $[P,Q]$ and $[P',Q']$ both in 2D Euclidean frames, with both having the same length, i.e. $d(P,Q) = d(P',Q') \gt 0$ , show that there exists exactly two motions such that $T(P) ...
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2answers
66 views

Find the diametre of the well

I'm stuck in solving this problem. Below, there is a well and $2$ sticks are thrown inside.The distance from the intersection of the rods to the bottom of the well is $1 m$ .What is the diametre ...
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1answer
42 views

Lines through vertices of $\triangle ABC$ and a point $Q$ meet opposite sides at $M$, $N$, $P$. When is $Q$ the orthocenter of $\triangle MNP$?

Consider a point $Q$ inside the $\triangle ABC$ triangle, and $M$, $N$, $P$ the intersections of $\overleftrightarrow{AQ}$, $\overleftrightarrow{BQ}$, $\overleftrightarrow{CQ}$ with respective sides $\...
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2answers
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A conjecture about the sum of the areas of three triangles built on the sides of any given triangle

Given any triangle $\triangle ABC$, and given one of its side, we can draw two lines perpendicular to that side passing through its two vertices. If we do this construction for each side, we obtain ...
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1answer
28 views

Find angle $ \angle AED $ in the following triangle. [duplicate]

Find angle $ \angle AED $ in the following triangle. In the above triangle we have : $CA=CB ,CE=DB=BA ,\angle ACB =20^° , \angle CAB=\angle CBA=80^°$ now find $ \angle AED $. I think if we draw ...
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0answers
33 views

Optimisation of sin, cosine angles of interior points of the triangle

If A, B, C are the angles such that $A+B+C=360^0$ then the minimum values of $SinA+sinB+sinC$, $cosA+cosB+cosC$ where A, B, C are the convex angles (less than 180) How to proceed to solve the problem,...
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0answers
7 views

Visualizing Pearson's Correlation Coefficient via Cosine Similarity

This is part of me trying to understand correlation via vector approach because I am not convinced with just empirical proof of final ratio working out between -1 and 1 as we needed. So was exploring ...
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3answers
72 views

The distances from a point to the corners of a rectangle are $6$, $7$, $9$, and (integer) $d$. Find $d$.

Christina is standing in a rectangular garden. Her distances from the corners of the garden are $6$ meters, $7$ meters, $9$ meters, and $d$ meters, where $d$ is an integer. How to find $d$? Can ...
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1answer
34 views

What does “Reflection along the subspace generated by v” means?

I got a problem which includes "Reflection along the subspace generated by $v$ in $\mathbf{R}^{n+1}$". I need some clarification, what does it mean? Does it mean reflection about the hyperplane $v^{\...
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0answers
17 views

Showing a group generated by reflections of the side of a hexagon is a discrete action on the Euclidean plane.

How would one go about showing that the action of the group generated by reflections with respect to the sides of a regular hexagon in the euclidean place, acts discretely on the euclidean plane.
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4answers
127 views

Norm inequality $\| x - y \| \cdot \| z \| \leq \| x - z \| \cdot \| y \| + \| z - y \| \cdot \| x \|$

Let $x$, $y$, $z$ be $3$ vectors in Euclidean space $V$. $\| x \|$ is norm of $x$ (length) How do you prove: $$\| x - y \| \cdot \|z\| \leq \| x - z\| \cdot \| y \| + \| z - y\| \cdot \| x \|?$$ I ...
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1answer
33 views

angle bisector and concurrency problem.

enter image description here In this figure, line BE is angle bisector of ∠ABC and some point X is on BE. If ∠AEB=∠FDB, then prove ∠EDF is right angle.
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2answers
65 views

Show that $OX=OY$.

Let $O$ the circumcenter of an acute triangle $ABC$. Let $\alpha $ the circle trough $A $ and $B $ tangent to $[AC] $, and $\beta $ the circle trough $A, C $ tangent to $[AB] $. A line trough $A $ ...
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0answers
24 views

Relation for angles between 3 unit vectors

Let there be $3$ unit vectors $\textbf{a}_1,\textbf{a}_2,\textbf{a}_3$ in $3\text{D}$. If we have $$\textbf{a}_1\cdot\textbf{a}_2=p$$ $$\textbf{a}_2\cdot\textbf{a}_3=q$$ $$\textbf{a}_1\cdot\textbf{a}...
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1answer
116 views

A simple conjecture (and a question) about three parabolas related to any triangle

Given any triangle, we can build three parabolas, each with focus on one vertex and with directrix the opposing side, as illustrated here: My first conjecture, likely trivial, is that, given any ...
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0answers
15 views

Degrees of freedom in hyperplane intuition?

I can describe an $n-1$ dimensional hyperplane in $R^n$ with a point and a single $n$ dimensional vector (namely, the normal vector). Similarly, I can describe a $1$ dimensional hyperplane (a line) ...
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1answer
28 views

Proof that the circumcenters of sub triangles forms a triangle congruent with the original triangle

Let $\triangle ABC$ be a triangle with orthocenter H and let $O_A, O_B, O_C$ be the circumcenters of triangles $\triangle BCH, \triangle CAH, \triangle ABH$, respectively. Prove that the $\triangle ...
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3answers
76 views

Proof of $tangent$ definition

I've seen a lot of "proofs" for this - but they all somehow revolve around the definition of $tan(x)$ being $\frac{opposite}{adjacent}$. I am coming from the standpoint of being unconvinced that this ...
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0answers
22 views

Can one state and prove that Euclidean space has genus $0$ in Hilbert's geometry?

Related to a more historically inclined question I'd like to ask, if the language and axioms of Hilbert's geometry do suffice to first state and then prove that in the standard model (Euclidean space) ...
3
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4answers
59 views

$P$ is a point on the angular bisector of $\angle A$. Show that $\frac{1}{AB}+\frac{1}{AC}$ doesn't depend on the line through $P$

The point $P$ is on the angular bisector of a given angle $\angle A$. A line $L$ is drawn through $P$ which intersects with the legs of the angle in $B$ and $C$. Show that $$\dfrac{1}{AB} + \dfrac{1}{...
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4answers
80 views

Determine all points $P$ on $\triangle ABC$ so that $|\triangle PAB| = |\triangle PBC|= |\triangle PAC|$.

Determine all points $P$ on $\triangle ABC$ so that $|\triangle PAB| = |\triangle PBC| = |\triangle PAC|$. Here, $|\triangle XYZ|$ denotes the area of $\triangle XYZ$. I've tried drawing it up, and $...
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0answers
59 views

Prove intersection of trapezium lies on perpendicular bisector of PQ

Let ABCD be a trapezium with AB and CD being parallel. Let P and Q be points lying on AD and BC respectively, such that $\angle$APB=$\angle$CPD and $\angle$AQB=$\angle$CQD. Let E be the point of ...
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2answers
102 views

How to dissect a rectangle to obtain an equal square?

There are several propositions and constructions in Euclid's Elements that relate to the squaring of a rectangle, e.g. Proposition II.5: If a straight line is cut into equal and unequal segments, ...
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2answers
71 views

Are $M, P, N $ collinear?

Let $\alpha $ a circle of diameters $AB $ and $\beta $ a circle tangent to $AB $ in $ C$ and tangent to $\alpha$ in $T $. Let $M\in \alpha $ and $N\in CB $ s.t. $MN\perp AB $ and $MN $ is tangent ...
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0answers
51 views

Show that $\angle AMC=\angle CMN $.

Let $\alpha $ a circle of diameters $AB $ and $\beta $ a circle tangent to $AB $ in $ C$ and tangent to $\alpha$ in $T $. Let $M\in \alpha $ and $N\in CB $ s.t. $MN\perp AB $ and $MN $ is tangent ...
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1answer
39 views

How to study Euclidean geometry from axioms?

I want to know if there's a good book or any other type of guide to study Euclidean geometry by only the 5 axioms in plane geometry and prove every other theorems from them?
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1answer
26 views

Show that the center of the circle $(ABC) $ is on $AE $.

Let $\triangle ABC $ and $D\in [BC] $ s.t. $\angle BAD=\angle DAC $. Let $BE\perp AD $ where $E $ is on the circle (ABD). Show that the center of the circle $(ABC) $ is on $AE $. I have no idea ...
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0answers
25 views

Impossibility of geometrical tools that solve equations of degree five

At Hartshorne, Geometry: Euclid and beyond (1997), p. 4, I read: Can from this and Abel-Ruffini's theorem be derived, that there are no (inherently physical) tools (like ordinary straightedge, marked ...